#495820

order	512
nilpotency class	4
isoclinism	isoclinic class of #33522
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	C₂²×D₈	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

714 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=i,d^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=di,f^c=fh,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403016);