#6368674

order	512
nilpotency class	3
isoclinism	isoclinic class of #6368558
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

1462 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:

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

Polycyclic presentation:

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

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306688, #72306694, #72306703
maximal normal subgroup of	#72306688, #72306694, #72306703

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6275870);