#495125

order	512
nilpotency class	4
isoclinism	isoclinic class of #33605
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1370 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)(11,20)(12,18)(13,27)(14,17)(15,29)(16,23)(19,21)(22,35)(24,34)(25,36)(26,33)(28,40)(30,38)(31,37)(32,39);(6,8)(9,11,21,37,15,22,28,38,14,24,26,39,12,25,13,23)(10,16,27,36,18,32,33,34,17,30,40,35,29,31,19,20);(1,2,3,4)(10,17)(16,30)(18,29)(19,33)(20,34)(27,40)(31,32)(35,36);(9,12,14,15)(10,18,17,29)(11,22,24,25)(13,26,28,21)(16,31,30,32)(19,27,33,40)(20,35,34,36)(23,37,38,39)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=fi,d^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402321);