#6911126

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1606 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fgi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,d^c=dfh,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6818322);