#7039737

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1317 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)(11,15)(12,16)(13,14)(17,18)(19,24)(20,32)(21,26)(22,36)(23,27)(25,34)(28,37)(29,39)(30,45)(31,40)(33,48)(35,43)(38,44)(41,47)(42,46);(1,3)(2,6)(4,7)(5,8)(10,14)(11,12)(17,19)(18,24)(20,33)(21,29)(22,30)(23,31)(25,42)(26,39)(27,40)(28,41)(32,48)(34,46)(35,44)(36,45)(37,47)(38,43);(9,10)(11,16)(12,15)(13,14)(17,20,23,21)(18,25,27,43)(19,29,31,33)(22,35,37,34)(24,38,40,42)(26,36,32,28)(30,46,47,44)(39,41,48,45);(1,2,5,4)(3,7,8,6)(9,11,13,12)(10,15,14,16)(17,21,23,20)(18,26,27,32)(19,29,31,33)(22,35,37,34)(24,39,40,48)(25,36,43,28)(30,44,47,46)(38,41,42,45);(1,4,5,2)(3,7,8,6)(9,12,13,11)(10,15,14,16)(18,27)(20,21)(24,40)(28,36)(29,33)(34,35)(41,45)(44,46)

Polycyclic presentation:

2^9|a^2=fhi,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,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, 6946933);