#6827043

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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	#30708	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1444 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)(11,14)(12,16)(13,15)(17,18,21,25)(19,27,29,23)(20,24,34,40)(22,26,33,39)(28,41,42,35)(30,38,32,36)(31,37,44,47)(43,48,45,46);(9,11,14,10)(12,16,13,15)(19,28)(20,31)(22,33)(23,35)(24,37)(26,39)(27,41)(29,42)(30,32)(34,44)(36,38)(40,47);(2,5)(6,8)(9,12,14,13)(10,15,11,16)(17,19,22,28,21,29,33,42)(18,23,26,35,25,27,39,41)(20,32,31,45,34,30,44,43)(24,38,37,48,40,36,47,46);(1,3,4,7)(2,6,5,8)(9,12,14,13)(10,15,11,16)(19,29)(23,27)(28,42)(30,32)(35,41)(36,38)(43,45)(46,48);(9,13,14,12)(10,15,11,16)(17,20,33,44,21,34,22,31)(18,24,39,47,25,40,26,37)(19,30,42,45,29,32,28,43)(23,36,41,48,27,38,35,46)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fh,d^2=gi,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=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, 6734239);