#6829890

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	#42806	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1677 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)(3,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18,23,22)(19,24,29,38)(20,25,33,32)(21,26,36,37)(27,30,39,46)(28,34,41,48)(31,40,35,42)(43,45,44,47);(1,3)(2,5)(4,8)(6,7)(17,19,23,29)(18,24,22,38)(20,30,33,46)(21,34,36,48)(25,39,32,27)(26,41,37,28)(31,44,35,43)(40,47,42,45);(10,13)(14,16)(17,20,18,25,23,33,22,32)(19,27,38,46,29,39,24,30)(21,35,26,42,36,31,37,40)(28,44,48,45,41,43,34,47);(2,6)(5,7)(21,36)(26,37)(28,41)(31,35)(34,48)(40,42)(43,44)(45,47);(3,8)(5,7)(9,11,12,15)(10,14,13,16)(17,21,18,26,23,36,22,37)(19,28,38,48,29,41,24,34)(20,31,25,40,33,35,32,42)(27,43,46,47,39,44,30,45)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=hi,e^2=ghi,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, 6737086);