#7002477

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

1403 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=i,d^2=fi,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, 6909673);