#498846

order	512
nilpotency class	4
isoclinism	isoclinic class of #498791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3155	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1630 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:

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

Polycyclic presentation:

2^9|a^2=i,e^2=hi,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,h^a=hi,c^b=cg,d^b=di,e^b=eh,f^b=fh,h^b=hi,e^c=eh,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, 406042);