#234291

order	512
nilpotency class	3
isoclinism	isoclinic class of #234289
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2252 conjugacy classes of subgroups; only the 204 normal subgroups are shown.

normal subgroups only

This group has 2252 conjugacy classes of subgroups — too many to draw. Only its 204 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)(3,5)(4,6)(7,8)(9,10,14,18)(11,19,22,32)(12,16,23,30)(13,25,27,17)(15,26,29,21)(20,33,34,40)(24,37,36,31)(28,38,39,35);(1,3)(2,5)(4,7)(6,8)(9,11,14,22)(10,15,18,29)(12,20,23,34)(13,26,27,21)(16,28,30,39)(17,32,25,19)(24,38,36,35)(31,40,37,33);(5,8)(9,12)(10,16)(11,20)(13,24)(14,23)(15,28)(17,31)(18,30)(19,33)(21,35)(22,34)(25,37)(26,38)(27,36)(29,39)(32,40);(12,23)(16,30)(20,34)(24,36)(28,39)(31,37)(33,40)(35,38)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 141487);