#6368747

order	512
nilpotency class	3
isoclinism	isoclinic class of #6368694
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1483 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=i,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,c^b=ch,d^b=df,e^b=ei,f^b=fh,e^c=eh,g^c=gi,e^d=ei,f^d=fh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306685, #72306694, #72306703
maximal normal subgroup of	#72306685, #72306694, #72306703

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6275943);