#6997075

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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1644 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)(17,18,22,25)(19,27,28,40)(20,24,31,39)(21,33,35,26)(23,34,36,30)(29,41,42,48)(32,45,44,38)(37,46,47,43);(1,2,4,5)(3,7,8,6)(9,10)(11,15)(12,14)(13,16)(19,28)(23,36)(27,40)(29,42)(30,34)(37,47)(41,48)(43,46);(9,11)(10,13)(12,16)(14,15)(17,19,22,28)(18,23,25,36)(20,29,31,42)(21,34,35,30)(24,37,39,47)(26,40,33,27)(32,46,44,43)(38,48,45,41);(1,3,4,8)(2,6,5,7)(11,16)(13,15)(17,20)(18,24)(19,29)(21,32)(22,31)(23,37)(25,39)(26,38)(27,41)(28,42)(30,43)(33,45)(34,46)(35,44)(36,47)(40,48);(10,14)(13,15)(18,25)(20,31)(23,36)(26,33)(27,40)(29,42)(32,44)(43,46)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^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=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, 6904271);