#6368762

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

1504 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=hi,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	#72306547, #72306687, #72306696, …
maximal normal subgroup of	#72306547, #72306687, #72306696, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6275958);