#7001951

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

1405 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=f,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=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, 6909147);