#511808

order	512
nilpotency class	4
isoclinism	isoclinic class of #511807
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#32898
commutator, Frattini, cosocle	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

2308 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 2308 conjugacy classes of subgroups — too many to draw. Only its 186 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2)(3,8)(4,17)(5,10)(6,23)(7,12)(9,20)(11,25)(13,36)(14,40)(15,34)(16,29)(18,46)(19,35)(21,32)(22,50)(24,31)(26,39)(27,41)(28,44)(30,47)(33,49)(37,59)(38,45)(42,54)(43,53)(48,57)(51,55)(52,64)(56,63)(58,61)(60,62);(1,3)(2,8)(4,13)(5,14)(6,15)(7,16)(9,26)(10,27)(11,34)(12,29)(17,45)(18,37)(19,38)(20,39)(21,41)(22,42)(23,28)(24,43)(25,44)(30,52)(31,53)(32,40)(33,56)(35,36)(46,59)(47,60)(48,61)(49,63)(50,54)(51,64)(55,62)(57,58);(1,4,7,19)(2,9,12,31)(3,13,16,38)(5,18,21,48)(6,24,25,20)(8,26,29,53)(10,30,32,55)(11,35,23,17)(14,37,41,61)(15,43,44,39)(22,51,49,47)(27,52,40,62)(28,45,34,36)(33,57,50,46)(42,64,63,60)(54,59,56,58);(1,5)(2,10)(3,14)(4,18)(6,22)(7,21)(8,27)(9,30)(11,33)(12,32)(13,37)(15,42)(16,41)(17,46)(19,48)(20,47)(23,50)(24,51)(25,49)(26,52)(28,54)(29,40)(31,55)(34,56)(35,57)(36,58)(38,61)(39,60)(43,64)(44,63)(45,59)(53,62)

Polycyclic presentation:

2^9|c^2=i,f^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,h^a=hi,f^b=fhi,g^b=gi,e^c=eh,f^c=fi,e^d=ei,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72209847, #72209850, #72249341, …
maximal normal subgroup of	#72209847, #72249341, #72249343
maximal quotient of	#72227660, #72227716, #72227718, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419004);