#6909402

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	#2615	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1487 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)(17,18,23,27)(19,28,29,40)(20,25,32,41)(21,26,22,37)(24,35,38,31)(30,45,33,48)(34,43,36,44)(39,46,42,47);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,14)(13,16)(19,29)(20,32)(24,38)(25,41)(28,40)(31,35)(34,36)(43,44);(3,8)(5,7)(9,11)(10,13)(12,16)(14,15)(17,19,22,31,23,29,21,35)(18,24,26,28,27,38,37,40)(20,33,34,46,32,30,36,47)(25,42,44,48,41,39,43,45);(2,6)(5,7)(11,16)(13,15)(17,20)(18,25)(19,30)(21,36)(22,34)(23,32)(24,39)(26,44)(27,41)(28,45)(29,33)(31,47)(35,46)(37,43)(38,42)(40,48);(10,14)(13,15)(17,21,23,22)(18,26,27,37)(19,31,29,35)(20,34,32,36)(24,40,38,28)(25,43,41,44)(30,46,33,47)(39,45,42,48)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=g,e^2=i,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,e^c=ei,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, 6816598);