#500202

order	512
nilpotency class	4
isoclinism	isoclinic class of #499855
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1002 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 1002 conjugacy classes of subgroups — too many to draw. Only its 134 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407398);