#416777

order	512
nilpotency class	3
isoclinism	isoclinic class of #394799
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	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

762 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=ef,b^2=eh,c^2=f,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,g^c=gi

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, 323973);