#6386526

order	512
nilpotency class	2
isoclinism	isoclinic class of #46350
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2350 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,h^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ci,d^b=dfi,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, 6293722);