#499004

order	512
nilpotency class	4
isoclinism	isoclinic class of #498791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3058	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1182 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406200);