#7471096

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309060
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ 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)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,26)(20,24)(21,30)(22,25)(23,28)(27,32)(29,31)(33,34,39,53,38,44,37,43)(35,40,47,56,49,58,48,57)(36,42,50,41,52,60,51,61)(45,62,64,54,46,55,63,59);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19,22,28)(18,23,25,26)(20,27,21,29)(24,31,30,32)(33,35,38,49)(34,40,44,58)(36,46,52,45)(37,48,39,47)(41,59,61,54)(42,55,60,62)(43,57,53,56)(50,63,51,64);(17,19)(18,23)(20,29)(21,27)(22,28)(24,32)(25,26)(30,31)(33,36,38,52)(34,41,44,61)(35,45,49,46)(37,50,39,51)(40,54,58,59)(42,43,60,53)(47,63,48,64)(55,56,62,57);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,20,22,21)(18,24,25,30)(19,27,28,29)(23,31,26,32)(33,36)(34,42)(35,46)(37,51)(38,52)(39,50)(40,55)(41,53)(43,61)(44,60)(45,49)(47,63)(48,64)(54,57)(56,59)(58,62);(2,8)(6,9)(7,16)(14,15)(17,21,22,20)(18,24,25,30)(19,27,28,29)(23,32,26,31)(33,37,38,39)(34,43,44,53)(35,47,49,48)(36,50,52,51)(40,56,58,57)(41,60,61,42)(45,63,46,64)(54,62,59,55)

Polycyclic presentation:

2^9|a^2=gh,b^2=fi,c^2=i,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfhi,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7378292);