#6803050

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1552 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)(17,18,23,26,22,27,21,36)(19,28,30,40,31,24,37,41)(20,32,34,44,35,43,38,25)(29,45,47,39,33,48,46,42);(2,5)(3,7)(9,10,12,13)(11,15,16,14)(17,19,22,31)(18,24,27,28)(20,33,35,29)(21,37,23,30)(25,42,44,39)(26,41,36,40)(32,45,43,48)(34,46,38,47);(1,3,4,7)(2,6,5,8)(10,13)(14,15)(17,20,22,35)(18,25,27,44)(19,29,31,33)(21,34,23,38)(24,39,28,42)(26,43,36,32)(30,46,37,47)(40,48,41,45);(2,5)(6,8)(17,21,22,23)(18,26,27,36)(19,30,31,37)(20,34,35,38)(24,40,28,41)(25,43,44,32)(29,46,33,47)(39,48,42,45);(9,11,12,16)(10,14,13,15)(18,27)(24,28)(25,44)(26,36)(32,43)(39,42)(40,41)(45,48)

Polycyclic presentation:

2^9|a^2=g,b^2=fhi,c^2=f,d^2=i,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,d^b=di,e^b=efh,g^c=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6710246);