#6999851

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×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1639 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^c=ei,g^c=gi,e^d=ei

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, 6907047);