#6992903

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

1588 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)(17,18,23,27)(19,24,29,39)(20,31,33,44)(21,26,22,36)(25,37,43,34)(28,45,32,48)(30,40,35,41)(38,46,42,47);(1,3,7,4)(2,5,8,6)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,32)(21,35)(22,30)(23,29)(25,42)(26,41)(27,39)(28,33)(31,48)(34,46)(36,40)(37,47)(38,43)(44,45);(1,4,7,3)(2,5,8,6)(11,16)(13,15)(17,20)(18,25)(19,28)(21,34)(22,37)(23,33)(24,38)(26,44)(27,43)(29,32)(30,46)(31,36)(35,47)(39,42)(40,45)(41,48);(1,5,7,6)(2,3,8,4)(10,14)(13,15)(19,29)(20,33)(24,39)(25,43)(30,35)(31,44)(34,37)(40,41);(1,6,7,5)(2,3,8,4)(17,21,23,22)(18,26,27,36)(19,30,29,35)(20,34,33,37)(24,40,39,41)(25,44,43,31)(28,46,32,47)(38,45,42,48)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fhi,d^2=fhi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,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, 6900099);