#6832142

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1469 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)(19,29)(20,25)(21,27)(22,26)(23,28)(24,31)(30,45)(32,43)(33,40)(34,41)(35,42)(36,44)(37,39)(38,47)(46,48);(1,3)(2,5)(4,8)(6,7)(17,19,22,31)(18,24,26,29)(20,33,34,37)(21,36,32,23)(25,39,41,40)(27,28,43,44)(30,46,47,35)(38,48,45,42);(10,13)(14,16)(17,20,23,35,22,34,36,46)(18,25,28,42,26,41,44,48)(19,30,32,33,31,47,21,37)(24,38,27,39,29,45,43,40);(2,6)(5,7)(18,26)(20,34)(24,29)(27,43)(28,44)(30,47)(33,37)(35,46);(3,8)(5,7)(9,11,12,15)(10,14,13,16)(17,21,36,31,22,32,23,19)(18,27,44,24,26,43,28,29)(20,33,46,30,34,37,35,47)(25,40,48,45,41,39,42,38)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,e^2=ghi,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, 6739338);