#6380732

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
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	C₂⁴	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1474 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6287928);