#420022

order	512
nilpotency class	3
isoclinism	isoclinic class of #417692
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1387 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 327218);