#7080110

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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1479 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=hi,d^2=gi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=ehi,f^b=fi,d^c=dhi,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, 6987306);