#7272476

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234485
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

1125 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7179672);