#422787

order	512
nilpotency class	3
isoclinism	isoclinic class of #417464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2023 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

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, 329983);