#498354

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
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 ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

672 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)(9,10,15,20,30,38,28,13,16,21,14,19,12,18,29,31)(11,22,25,36,23,40,32,17,27,35,24,34,26,39,33,37);(5,7,8,6)(9,11,16,27)(10,17,21,37)(12,23,30,26)(13,22,31,35)(14,32,15,33)(18,34,38,36)(19,39,20,40)(24,29,25,28);(2,4)(9,12,16,30)(10,18,21,38)(11,23,27,26)(13,20,31,19)(14,28,15,29)(17,34,37,36)(22,40,35,39)(24,33,25,32);(9,13,28,18,12,20,15,21,16,31,29,38,30,19,14,10)(11,17,32,39,26,36,25,35,27,37,33,40,23,34,24,22)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405550);