#6997072

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1661 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)(6,7)(9,10)(11,13)(12,14)(15,16)(17,18,23,29)(19,30,33,25)(20,34,21,27)(22,28,24,39)(26,37,44,35)(31,45,36,41)(32,43,38,42)(40,48,46,47);(1,2,4,5)(3,7,8,6)(9,10)(11,15)(12,14)(13,16)(17,19,23,33)(18,25,29,30)(20,31,21,36)(22,38,24,32)(26,40,44,46)(27,45,34,41)(28,43,39,42)(35,48,37,47);(9,11)(10,13)(12,16)(14,15)(17,20)(18,26)(19,31)(21,23)(22,35)(24,37)(25,40)(27,28)(29,44)(30,46)(32,47)(33,36)(34,39)(38,48)(41,42)(43,45);(1,3,4,8)(2,6,5,7)(11,16)(13,15)(17,21,23,20)(18,27,29,34)(19,31,33,36)(22,35,24,37)(25,41,30,45)(26,39,44,28)(32,48,38,47)(40,43,46,42);(10,14)(13,15)(17,22,23,24)(18,28,29,39)(19,32,33,38)(20,35,21,37)(25,42,30,43)(26,27,44,34)(31,47,36,48)(40,41,46,45)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6904268);