#6910225

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#37682	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1846 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=fi,c^2=g,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,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, 6817421);