#493814

order	512
nilpotency class	4
isoclinism	isoclinic class of #33607
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	C₂²×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

998 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,e^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=di,d^c=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, 401010);