#6823400

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	#42760	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1406 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=fgi,e^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,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, 6730596);