#7002526

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1378 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)(17,18,21,26)(19,28,31,23)(20,25,33,43)(22,36,35,27)(24,37,41,34)(29,39,30,42)(32,40,48,45)(38,46,44,47);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,23)(20,29)(21,31)(22,32)(24,38)(25,42)(26,28)(27,40)(30,33)(34,46)(35,48)(36,45)(37,47)(39,43)(41,44);(9,11)(10,13)(12,16)(14,15)(17,20,21,33)(18,24,26,41)(19,29,31,30)(22,37,35,34)(23,38,28,44)(25,27,43,36)(32,47,48,46)(39,45,42,40);(1,2,5,4)(3,7,8,6)(11,16)(13,15)(17,20)(18,25)(19,30)(21,33)(22,34)(23,39)(24,36)(26,43)(27,41)(28,42)(29,31)(32,47)(35,37)(38,40)(44,45)(46,48);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(19,31)(20,33)(23,28)(24,41)(25,43)(32,48)(34,37)(40,45)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=i,d^2=fi,e^2=fi,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,e^c=ei,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, 6909722);