#6830620

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	#42674	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1484 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)(17,18,21,23)(19,24,29,36)(20,30,33,25)(22,26,35,37)(27,40,44,38)(28,42,32,39)(31,41,34,45)(43,47,46,48);(9,11,12,16)(10,13,14,15)(18,23)(19,27)(20,31)(22,35)(24,38)(25,41)(29,44)(30,45)(33,34)(36,40)(39,42)(43,46);(2,5)(6,8)(10,14)(11,16)(17,19,22,27,21,29,35,44)(18,24,26,40,23,36,37,38)(20,32,34,46,33,28,31,43)(25,42,41,47,30,39,45,48);(10,14)(13,15)(18,23)(20,33)(24,36)(26,37)(28,32)(31,34)(38,40)(43,46);(1,3,4,7)(2,6,5,8)(17,20,22,34,21,33,35,31)(18,25,26,41,23,30,37,45)(19,28,27,43,29,32,44,46)(24,39,40,48,36,42,38,47)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=fhi,e^2=gh,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, 6737816);