#494862

order	512
nilpotency class	4
isoclinism	isoclinic class of #33522
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×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	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

826 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=e,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,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, 402058);