#511795

order	512
nilpotency class	4
isoclinism	isoclinic class of #511793
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

2304 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 2304 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)(3,13)(4,9)(5,10)(6,11)(7,12)(8,16)(14,41)(15,45)(17,40)(18,29)(19,30)(20,31)(21,32)(22,33)(23,34)(24,36)(25,37)(26,43)(27,47)(28,50)(35,53)(38,44)(39,48)(42,60)(46,61)(49,57)(51,58)(52,59)(54,62)(55,64)(56,63);(1,3)(2,8)(4,14)(5,15)(6,16)(7,17)(9,26)(10,27)(11,13)(12,28)(18,42)(19,43)(20,44)(21,46)(22,47)(23,48)(24,49)(25,50)(29,54)(30,41)(31,55)(32,56)(33,45)(34,57)(35,38)(36,39)(37,40)(51,62)(52,63)(53,64)(58,60)(59,61);(1,4,7,20)(2,9,12,31)(3,14,17,44)(5,21,23,18)(6,19,25,53)(8,26,28,55)(10,32,34,29)(11,35,37,30)(13,38,40,41)(15,46,48,42)(16,43,50,64)(22,52,24,51)(27,56,57,54)(33,58,36,59)(39,61,45,60)(47,63,49,62);(1,5,7,23)(2,10,12,34)(3,15,17,48)(4,18,20,21)(6,24,25,22)(8,27,28,57)(9,29,31,32)(11,36,37,33)(13,39,40,45)(14,42,44,46)(16,49,50,47)(19,52,53,51)(26,54,55,56)(30,59,35,58)(38,60,41,61)(43,63,64,62)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256592, #72256595, #72303197
maximal normal subgroup of	#72256592, #72256595, #72303197
maximal quotient of	#72256600, #72261275, #72261301, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 418991);