#6986021

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907909
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 ◁ #2295 ◁ 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fh,c^2=fg,d^2=fi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfhi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6893217);