#477018

order	512
nilpotency class	3
isoclinism	isoclinic class of #476092
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1587 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 384214);