#6989716

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	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1499 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=g,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=dfhi,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, 6896912);