#6966274

order	512
nilpotency class	3
isoclinism	isoclinic class of #6885065
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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 ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1433 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)(17,18,23,26,22,28,21,27)(19,24,30,41,32,39,31,38)(20,33,34,25,36,42,35,40)(29,43,46,37,45,47,44,48);(1,3,4,8)(2,5,6,7)(10,13)(11,15)(17,19,22,32)(18,24,28,39)(20,29,36,45)(21,31,23,30)(25,37,40,48)(26,41,27,38)(33,43,42,47)(34,46,35,44);(3,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,20)(18,25)(19,29)(21,34)(22,36)(23,35)(24,37)(26,33)(27,42)(28,40)(30,44)(31,46)(32,45)(38,47)(39,48)(41,43);(2,6)(5,7)(10,13)(14,16)(17,21,22,23)(18,26,28,27)(19,30,32,31)(20,34,36,35)(24,38,39,41)(25,33,40,42)(29,44,45,46)(37,47,48,43);(9,11,12,15)(10,14,13,16)

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,d^2=i,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,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, 6873470);