#7486896

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309080
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

1128 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)(19,22)(20,24)(21,25)(23,26)(27,30)(28,31)(29,32)(33,34,39,41,37,43,38,44)(35,45,53,57,47,40,48,55)(36,49,54,60,51,42,52,58)(46,56,62,63,50,59,61,64);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19,21,26)(18,22,25,23)(20,28,29,27)(24,31,32,30)(33,35)(34,40)(36,50)(37,47)(38,48)(39,53)(41,55)(42,59)(43,45)(44,57)(46,51)(49,56)(52,62)(54,61)(58,64)(60,63);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,29)(30,32)(34,41)(38,39)(40,55)(42,58)(43,44)(45,57)(48,53)(49,60)(52,54)(56,63)(59,64)(61,62);(3,11)(6,9)(10,16)(13,15)(19,26)(22,23)(27,28)(30,31)(33,36)(34,42)(35,46)(37,51)(38,52)(39,54)(40,56)(41,58)(43,49)(44,60)(45,59)(47,50)(48,61)(53,62)(55,63)(57,64);(2,8)(6,9)(7,14)(13,15)(17,20,21,29)(18,24,25,32)(19,27,26,28)(22,30,23,31)(35,47)(36,51)(40,45)(42,49)(48,53)(52,54)(55,57)(58,60)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7394092);