#7031034

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
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

1366 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)(17,18,21,23)(19,27,28,41)(20,25,31,36)(22,34,33,26)(24,37,35,30)(29,40,42,48)(32,45,44,39)(38,47,46,43);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,14)(13,16)(18,23)(19,28)(20,31)(26,34)(30,37)(32,44)(38,46)(40,48);(9,11)(10,13)(12,16)(14,15)(17,19,22,30,21,28,33,37)(18,24,26,27,23,35,34,41)(20,29,32,43,31,42,44,47)(25,38,39,40,36,46,45,48);(1,2,5,4)(3,7,8,6)(11,16)(13,15)(17,20)(18,25)(19,29)(21,31)(22,32)(23,36)(24,38)(26,39)(27,40)(28,42)(30,43)(33,44)(34,45)(35,46)(37,47)(41,48);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(18,23)(20,31)(24,35)(26,34)(27,41)(29,42)(32,44)(43,47)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=g,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^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6938230);