#7184427

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1411 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)(3,5)(4,6)(7,8)(10,13)(11,15)(17,18,21,27,22,26,38,41)(19,28,31,40,30,24,37,23)(20,32,35,46,36,25,43,45)(29,42,33,39,34,44,47,48);(1,3,4,7)(2,5,6,8)(10,13)(11,15)(17,19,22,30)(18,23,26,40)(20,33,36,47)(21,37,38,31)(24,41,28,27)(25,44,32,42)(29,43,34,35)(39,46,48,45);(2,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,34)(21,31)(22,30)(23,27)(25,44)(26,28)(29,36)(32,42)(33,43)(35,47)(37,38)(39,46)(40,41)(45,48);(10,13)(14,16)(17,20,21,35,22,36,38,43)(18,25,27,45,26,32,41,46)(19,29,31,33,30,34,37,47)(23,39,28,44,40,48,24,42);(9,11,12,15)(10,14,13,16)(18,26)(19,30)(25,32)(27,41)(29,34)(31,37)(33,47)(45,46)

Polycyclic presentation:

2^9|a^2=f,b^2=ghi,d^2=f,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=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, 7091623);