#6930880

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	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1715 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=fhi,e^2=i,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,e^c=ei,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, 6838076);