#6912698

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	#29607	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1654 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,5,6)(3,7,8,4)(17,18)(19,27)(20,31)(21,25)(22,35)(23,29)(24,33)(26,34)(28,41)(30,37)(32,46)(36,48)(38,47)(39,43)(40,44)(42,45);(2,6)(4,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,36)(25,27)(26,37)(31,45)(32,42)(33,43)(34,44)(35,40)(38,39)(41,46)(47,48);(1,3)(2,7)(4,6)(5,8)(9,11)(10,13)(12,16)(14,15)(17,20,22,33,21,32,34,47)(18,24,26,31,25,38,35,46)(19,28,30,43,29,42,44,48)(23,36,37,45,27,39,40,41);(2,6)(3,8)(11,16)(13,15)(18,25)(19,29)(20,32)(26,35)(30,44)(33,47)(36,39)(41,45);(1,4,5,7)(2,3,6,8)(10,14)(13,15)

Polycyclic presentation:

2^9|a^2=fi,c^2=g,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,d^c=dfh

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, 6819894);