#190118

order	512
nilpotency class	3
isoclinism	isoclinic class of #190116
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#381
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

1806 conjugacy classes of subgroups; only the 250 normal subgroups are shown.

normal subgroups only

This group has 1806 conjugacy classes of subgroups — too many to draw. Only its 250 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)(3,7,8,5)(4,6)(9,10)(11,19,22,15)(12,23)(13,17)(14,18)(16,25)(20,28,36,33)(21,34,38,29)(24,31)(26,30)(27,32)(35,39,37,40);(9,11)(10,15)(12,20)(13,21)(14,22)(16,28)(17,29)(18,19)(23,33)(24,35)(25,36)(26,37)(27,38)(30,39)(31,40)(32,34);(1,3)(2,5)(4,8)(6,7)(9,12)(10,16,18,23)(11,20)(13,26)(14,25)(15,28,19,33)(17,31,32,30)(21,37)(22,36)(24,27)(29,40,34,39)(35,38);(9,13,14,27)(10,17,18,32)(11,21,22,38)(12,24,25,26)(15,29,19,34)(16,30,23,31)(20,35,36,37)(28,39,33,40)

Polycyclic presentation:

2^9|a^2=g,c^2=e,d^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^b=gh,d^c=dh,f^c=fh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 97314);