#7487720

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1131 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:

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=fhi,c^2=f,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,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

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7394916);