#6966339

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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1446 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)(17,18,23,36,22,27,21,26)(19,24,29,38,31,39,30,40)(20,32,33,25,35,41,34,42)(28,43,46,37,45,48,44,47);(1,3,4,8)(2,5,6,7)(10,13)(11,15)(17,19,22,31)(18,24,27,39)(20,28,35,45)(21,30,23,29)(25,37,42,47)(26,40,36,38)(32,43,41,48)(33,46,34,44);(3,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,20)(18,25)(19,28)(21,33)(22,35)(23,34)(24,37)(26,41)(27,42)(29,44)(30,46)(31,45)(32,36)(38,43)(39,47)(40,48);(2,6)(5,7)(10,13)(14,16)(17,21,22,23)(18,26,27,36)(19,29,31,30)(20,33,35,34)(24,38,39,40)(25,41,42,32)(28,44,45,46)(37,43,47,48);(9,11,12,15)(10,14,13,16)(18,27)(24,39)(25,42)(26,36)(32,41)(37,47)(38,40)(43,48)

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=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, 6873535);