#6812825

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1282 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)(11,13)(12,14)(15,16)(17,18,21,23)(19,27,30,24)(20,25,33,36)(22,26,35,37)(28,38,44,40)(29,42,32,39)(31,45,34,41)(43,47,46,48);(1,2)(3,6)(4,5)(7,8)(9,11,12,15)(10,13,14,16)(18,23)(19,28)(20,31)(22,35)(24,38)(25,41)(27,40)(30,44)(33,34)(36,45)(39,42)(43,46);(10,14)(13,16)(17,19,22,28,21,30,35,44)(18,24,26,40,23,27,37,38)(20,32,34,46,33,29,31,43)(25,42,41,47,36,39,45,48);(1,3,4,7)(2,6,5,8)(18,23)(20,33)(24,27)(26,37)(29,32)(31,34)(38,40)(43,46);(2,5)(6,8)(17,20,22,34,21,33,35,31)(18,25,26,41,23,36,37,45)(19,29,28,43,30,32,44,46)(24,39,40,48,27,42,38,47)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=h,d^2=fi,e^2=h,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, 6720021);