#6990200

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1349 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)(3,6)(4,8)(5,7)(9,10,12,14)(11,15,16,13)(17,18,23,27)(19,28,32,43)(20,25,34,41)(21,26,22,36)(24,30,38,31)(29,42,45,47)(33,39,35,40)(37,46,48,44);(1,2,6,3)(4,7,5,8)(9,11,12,16)(10,13,14,15);(1,3,6,2)(4,7,5,8)(11,16)(13,15)(17,19,21,30,23,32,22,31)(18,24,36,43,27,38,26,28)(20,29,35,46,34,45,33,44)(25,37,39,47,41,48,40,42);(1,4,6,5)(2,7,3,8)(10,14)(13,15)(17,20)(18,25)(19,29)(21,35)(22,33)(23,34)(24,37)(26,40)(27,41)(28,42)(30,46)(31,44)(32,45)(36,39)(38,48)(43,47);(1,5,6,4)(2,7,3,8)(17,21,23,22)(18,26,27,36)(19,30,32,31)(20,33,34,35)(24,28,38,43)(25,39,41,40)(29,44,45,46)(37,47,48,42)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6897396);