#7083194

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

1379 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)(19,26,31,40)(20,32,35,25)(21,27,22,38)(24,36,41,30)(29,43,33,46)(34,45,37,44)(39,48,42,47);(9,10)(11,15)(12,14)(13,16)(17,19,21,36,23,31,22,30)(18,24,38,40,28,41,27,26)(20,33,37,48,35,29,34,47)(25,42,44,46,32,39,45,43);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,29)(21,37)(22,34)(23,35)(24,39)(26,46)(27,45)(28,32)(30,48)(31,33)(36,47)(38,44)(40,43)(41,42);(1,4,5,3)(2,6,7,8)(11,16)(13,15)(17,19)(18,26)(20,33)(21,30)(22,36)(23,31)(24,27)(25,43)(28,40)(29,35)(32,46)(34,48)(37,47)(38,41)(39,44)(42,45);(10,14)(13,15)(17,21,23,22)(18,27,28,38)(19,30,31,36)(20,34,35,37)(24,40,41,26)(25,44,32,45)(29,47,33,48)(39,46,42,43)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=gh,d^2=gi,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, 6990390);