#7188488

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1247 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)(3,6)(4,8)(5,7)(10,14)(12,15)(17,18,22,25)(19,27,30,42)(20,24,33,41)(21,35,36,26)(23,29,38,31)(28,43,32,46)(34,48,47,40)(37,45,39,44);(1,3,5,4)(2,6,7,8)(9,10,13,14)(11,12,16,15)(17,19)(18,23)(20,32)(21,29)(22,30)(24,39)(25,38)(26,42)(27,35)(28,33)(31,36)(34,44)(37,41)(40,43)(45,47)(46,48);(2,7)(6,8)(9,11)(10,15)(12,14)(13,16)(17,20)(18,24)(19,28)(21,34)(22,33)(23,37)(25,41)(26,40)(27,43)(29,45)(30,32)(31,44)(35,48)(36,47)(38,39)(42,46);(1,4,5,3)(2,8,7,6)(10,14)(11,16)(18,25)(19,29)(20,33)(21,36)(23,27)(28,44)(30,31)(32,45)(37,46)(38,42)(39,43)(40,48);(9,12,13,15)(10,11,14,16)(18,25)(19,30)(20,33)(26,35)(29,31)(34,47)(37,39)(43,46)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,d^2=g,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,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, 7095684);