#6828852

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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	#42906	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1449 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)(11,15)(12,14)(13,16)(17,18)(19,29)(20,25)(21,35)(22,26)(23,28)(24,31)(27,37)(30,39)(32,40)(33,42)(34,43)(36,41)(38,48)(44,47)(45,46);(9,11)(10,13)(12,16)(14,15)(17,19,22,31)(18,24,26,29)(20,33,32,23)(21,34,37,36)(25,28,40,42)(27,43,35,41)(30,38,46,47)(39,44,45,48);(2,5)(6,8)(10,14)(11,16)(17,20,23,19,22,32,33,31)(18,25,28,29,26,40,42,24)(21,36,38,46,37,34,47,30)(27,43,44,39,35,41,48,45);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(18,26)(21,37)(24,29)(25,40)(28,42)(30,46)(34,36)(38,47);(17,21,23,38,22,37,33,47)(18,27,28,44,26,35,42,48)(19,30,32,36,31,46,20,34)(24,39,25,41,29,45,40,43)

Polycyclic presentation:

2^9|b^2=i,c^2=fhi,d^2=fgi,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6736048);