#6832303

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1231 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)(9,10,12,13)(11,14,15,16)(17,18)(19,25)(20,26)(21,27)(22,39)(23,29)(24,30)(28,32)(31,40)(33,41)(34,42)(35,43)(36,47)(37,44)(38,45)(46,48);(1,3,4,8)(2,5,6,7)(17,19)(18,25)(20,34)(21,32)(22,23)(24,33)(26,42)(27,28)(29,39)(30,41)(31,37)(35,46)(36,38)(40,44)(43,48)(45,47);(10,13)(14,16)(17,20,23,37,24,38,21,35)(18,26,29,44,30,45,27,43)(19,31,32,34,33,46,22,36)(25,40,28,42,41,48,39,47);(2,6)(5,7)(17,21,24,23)(18,27,30,29)(19,32,33,22)(20,35,38,37)(25,28,41,39)(26,43,45,44)(31,34,46,36)(40,42,48,47);(3,8)(5,7)(9,11,12,15)(10,14,13,16)(17,22,21,33,24,32,23,19)(18,28,27,25,30,39,29,41)(20,36,35,46,38,34,37,31)(26,42,43,40,45,47,44,48)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=h,d^2=i,e^2=ghi,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^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, 6739499);