#411178

order	512
nilpotency class	3
isoclinism	isoclinic class of #395166
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

972 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=ghi,c^2=g,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318374);