#6966607

order	512
nilpotency class	3
isoclinism	isoclinic class of #6885129
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

1395 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=i,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,d^c=dfh,g^c=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6873803);