#6948069

order	512
nilpotency class	3
isoclinism	isoclinic class of #6885129
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 ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1135 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)(3,11)(4,5)(6,9)(7,10)(8,14)(12,15)(13,16)(17,18)(19,22)(20,24)(21,23)(25,29)(26,28)(27,30)(31,32)(33,34,39,51,38,42,37,43)(35,40,52,62,45,54,46,55)(36,47,50,41,49,57,48,56)(44,58,61,53,59,64,60,63);(1,3)(2,8)(4,12)(5,13)(6,15)(7,14)(9,16)(10,11)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32)(33,35,38,45)(34,40,42,54)(36,44,49,59)(37,46,39,52)(41,53,56,63)(43,55,51,62)(47,58,57,64)(48,60,50,61);(1,4,7,9)(2,5,10,6)(3,12,14,16)(8,13,11,15)(18,23)(19,25)(20,27)(28,32)(33,36,38,49)(34,41,42,56)(35,44,45,59)(37,50,39,48)(40,53,54,63)(43,57,51,47)(46,61,52,60)(55,64,62,58);(1,5,7,6)(2,9,10,4)(3,13,14,15)(8,16,11,12)(17,20,21,27)(18,24,23,30)(19,26,25,31)(22,28,29,32)(35,45)(40,54)(44,59)(46,52)(53,63)(55,62)(58,64)(60,61);(1,6,7,5)(2,9,10,4)(3,13,14,15)(8,12,11,16)(19,25)(22,29)(26,31)(28,32)(34,42)(40,54)(41,56)(43,51)(47,57)(53,63)(55,62)(58,64)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=fi,d^2=fh,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi

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, 6855265);