#6801519

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667980
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	C₂×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1562 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)(17,18,22,27,23,28,21,26)(19,24,30,39,31,40,36,43)(20,32,34,42,35,44,37,25)(29,45,47,41,33,48,46,38);(1,3)(2,5)(4,8)(6,7)(9,10,12,13)(11,15,16,14)(17,19,23,31)(18,24,28,40)(20,33,35,29)(21,36,22,30)(25,41,42,38)(26,43,27,39)(32,48,44,45)(34,46,37,47);(10,13)(14,15)(17,20,23,35)(18,25,28,42)(19,29,31,33)(21,34,22,37)(24,38,40,41)(26,32,27,44)(30,46,36,47)(39,48,43,45);(2,6)(5,7)(17,21,23,22)(18,26,28,27)(19,30,31,36)(20,34,35,37)(24,39,40,43)(25,32,42,44)(29,46,33,47)(38,48,41,45);(3,8)(5,7)(9,11,12,16)(10,14,13,15)(19,31)(24,40)(29,33)(30,36)(38,41)(39,43)(45,48)(46,47)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,g^c=gi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6708715);