#511832

order	512
nilpotency class	4
isoclinism	isoclinic class of #511831
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	C₂×#41	C₂⁴
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

2277 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 2277 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)(3,17)(4,22)(5,13)(6,28)(7,15)(8,14)(9,16)(10,19)(11,25)(12,27)(18,46)(20,53)(21,42)(23,38)(24,57)(26,44)(29,37)(30,41)(31,34)(32,40)(33,59)(35,52)(36,51)(39,56)(43,55)(45,48)(47,54)(49,61)(50,60)(58,63)(62,64);(1,3)(2,10)(4,18)(5,19)(6,29)(7,31)(8,20)(9,21)(11,33)(12,34)(13,42)(14,35)(15,36)(16,37)(17,28)(22,54)(23,47)(24,58)(25,49)(26,50)(27,52)(30,51)(32,48)(38,60)(39,45)(40,62)(41,53)(43,61)(44,63)(46,57)(55,64)(56,59);(1,4)(2,11)(3,18)(5,23)(6,24)(7,32)(8,25)(9,26)(10,33)(12,38)(13,43)(14,44)(15,22)(16,40)(17,45)(19,47)(20,49)(21,50)(27,56)(28,39)(29,58)(30,55)(31,48)(34,60)(35,63)(36,54)(37,62)(41,57)(42,61)(46,53)(51,64)(52,59);(1,5,9,6)(2,12,16,41)(3,19,21,29)(4,23,26,24)(7,27,8,30)(10,34,37,53)(11,38,40,57)(13,15,28,14)(17,35,42,36)(18,47,50,58)(20,51,31,52)(22,39,44,43)(25,55,32,56)(33,60,62,46)(45,63,61,54)(48,59,49,64)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419028);