#6378510

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	#392	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1358 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,25,22,26,33,40)(19,23,29,36,27,34,41,45)(20,24,32,39,31,38,44,48)(28,35,42,46,30,37,43,47);(2,5)(3,7)(9,11,12,15)(10,13,14,16)(19,27)(23,34)(28,30)(29,41)(35,37)(36,45)(42,43)(46,47);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(17,19)(18,23)(20,30)(21,29)(22,27)(24,37)(25,36)(26,34)(28,31)(32,43)(33,41)(35,38)(39,47)(40,45)(42,44)(46,48);(2,5)(6,8)(19,27)(20,31)(23,34)(24,38)(29,41)(32,44)(36,45)(39,48);(17,20,21,32,22,31,33,44)(18,24,25,39,26,38,40,48)(19,28,29,42,27,30,41,43)(23,35,36,46,34,37,45,47)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=cfi,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6285706);