#495872

order	512
nilpotency class	4
isoclinism	isoclinic class of #33607
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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	C₂×#466	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

704 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)(5,6,8,7)(10,15)(12,24)(13,25)(16,26)(17,27)(18,31)(19,30)(20,35)(22,33)(23,38)(28,29)(32,37)(34,40)(36,39);(9,10,16,31,13,18,12,15,14,19,24,29,25,28,26,30)(11,20,32,39,23,36,22,35,21,17,33,40,38,34,37,27);(2,4)(9,11)(10,17)(12,22)(13,23)(14,21)(15,27)(16,32)(18,34)(19,20)(24,33)(25,38)(26,37)(28,36)(29,39)(30,35)(31,40);(5,7,8,6)(11,21)(17,20)(22,37)(23,38)(27,35)(32,33)(34,36)(39,40)

Polycyclic presentation:

2^9|a^2=g,b^2=ehi,d^2=g,e^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,c^b=ci,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, 403068);