#6992922

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 ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1601 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)(9,10)(11,15)(12,14)(13,16)(17,18,22,26)(19,29,31,24)(20,33,34,42)(21,28,23,27)(25,37,40,35)(30,44,46,48)(32,39,36,41)(38,47,43,45);(1,3,7,4)(2,5,8,6)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,30)(21,36)(22,31)(23,32)(25,38)(26,29)(27,41)(28,39)(33,48)(34,46)(35,47)(37,45)(40,43)(42,44);(1,4,7,3)(2,5,8,6)(11,16)(13,15)(17,20,22,34)(18,25,26,40)(19,30,31,46)(21,35,23,37)(24,38,29,43)(27,33,28,42)(32,45,36,47)(39,44,41,48);(1,5,7,6)(2,3,8,4)(10,14)(13,15)(18,26)(19,31)(20,34)(27,28)(32,36)(35,37)(38,43)(44,48);(1,6,7,5)(2,3,8,4)(17,21,22,23)(18,27,26,28)(19,32,31,36)(20,35,34,37)(24,39,29,41)(25,33,40,42)(30,45,46,47)(38,44,43,48)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=fh,d^2=fhi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,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, 6900118);