#7272231

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234437
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1151 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)(3,10)(4,8)(5,14)(6,9)(7,13)(11,16)(12,15)(17,18,21,24)(19,22,26,30)(20,23,28,32)(25,29,27,31)(33,34,38,43)(35,44,39,40)(36,47,50,53)(37,42,52,58)(41,45,56,48)(46,57,51,54)(49,60,62,63)(55,59,64,61);(1,3)(2,7)(4,11)(5,12)(6,13)(8,15)(9,10)(14,16)(17,19,21,26)(18,22,24,30)(20,27,28,25)(23,31,32,29)(33,35,38,39)(34,40,43,44)(36,48,50,45)(37,51,52,46)(41,47,56,53)(42,57,58,54)(49,59,62,61)(55,63,64,60);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,10,16)(17,20)(18,23)(19,25)(21,28)(22,29)(24,32)(26,27)(30,31)(33,36,35,45,38,50,39,48)(34,41,40,53,43,56,44,47)(37,49,46,59,52,62,51,61)(42,55,54,63,58,64,57,60);(1,4,6,5)(2,8,9,14)(3,12,13,11)(7,16,10,15)(20,28)(23,32)(25,27)(29,31)(33,37)(34,42)(35,46)(36,49)(38,52)(39,51)(40,54)(41,55)(43,58)(44,57)(45,59)(47,60)(48,61)(50,62)(53,63)(56,64);(1,5,6,4)(2,8,9,14)(3,12,13,11)(7,15,10,16)(19,26)(22,30)(25,27)(29,31)(34,43)(37,52)(40,44)(41,56)(46,51)(47,53)(49,62)(59,61)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fg,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7179427);