#7048309

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996375
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1309 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)(10,14)(12,15)(17,18,22,27)(19,23,30,26)(20,31,33,35)(21,25,34,40)(24,37,39,28)(29,36,43,41)(32,44,45,46)(38,47,48,42);(1,3)(2,6)(4,7)(5,8)(9,10)(11,12)(13,14)(15,16)(17,19,22,30)(18,23,27,26)(20,28,33,37)(21,29,34,43)(24,35,39,31)(25,36,40,41)(32,42,45,47)(38,46,48,44);(1,2,5,4)(3,6,8,7)(9,11,13,16)(10,15,14,12)(17,20,19,28,22,33,30,37)(18,24,26,31,27,39,23,35)(21,32,29,42,34,45,43,47)(25,38,41,44,40,48,36,46);(3,8)(6,7)(10,14)(11,16)(17,21,22,34)(18,25,27,40)(19,29,30,43)(20,32,33,45)(23,36,26,41)(24,38,39,48)(28,42,37,47)(31,44,35,46);(1,4,5,2)(3,7,8,6)(9,12,13,15)(10,16,14,11)(21,34)(25,40)(29,43)(32,45)(36,41)(38,48)(42,47)(44,46)

Polycyclic presentation:

2^9|a^2=fhi,b^2=i,c^2=fg,d^2=i,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=eh,d^c=dh,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6955505);