#6364211

order	512
nilpotency class	3
isoclinism	isoclinic class of #6363930
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

1472 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)(3,13,14,27)(4,9,19,26)(5,11,6,10)(8,20,22,16)(15,29,30,24)(17,32,31,23)(18,28,21,25)(33,34)(35,40)(36,47)(37,42)(38,51)(39,43)(41,48)(44,59)(45,54)(46,56)(49,57)(50,55)(52,58)(53,61)(60,63)(62,64);(1,3)(2,8)(4,17)(5,16)(6,20)(7,14)(9,24)(10,13)(11,27)(12,22)(15,18)(19,31)(21,30)(23,25)(26,29)(28,32)(33,35,39,46)(34,40,43,56)(36,44,49,62)(37,50,38,45)(41,53,58,63)(42,55,51,54)(47,59,57,64)(48,61,52,60);(3,14)(8,22)(13,27)(15,30)(16,20)(17,31)(23,32)(24,29)(33,36)(34,41)(35,44)(37,48)(38,52)(39,49)(40,53)(42,57)(43,58)(45,60)(46,62)(47,51)(50,61)(54,59)(55,64)(56,63);(1,4,7,19)(2,9,12,26)(3,15,14,30)(5,18,6,21)(8,23,22,32)(10,25,11,28)(13,29,27,24)(16,31,20,17);(1,5,7,6)(2,10,12,11)(3,16,14,20)(4,18,19,21)(8,13,22,27)(9,25,26,28)(15,31,30,17)(23,29,32,24)(33,37,39,38)(34,42,43,51)(35,45,46,50)(36,48,49,52)(40,54,56,55)(41,57,58,47)(44,60,62,61)(53,59,63,64)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306963, #72306977, #72307019
maximal normal subgroup of	#72306963, #72306977, #72307019

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6271407);