#7083212

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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

1384 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,c^2=ghi,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,e^c=ei,f^d=fi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6990408);