#7191146

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1432 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=ghi,c^2=g,d^2=fgi,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,e^d=ei

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, 7098342);