#251386

order	512
nilpotency class	3
isoclinism	isoclinic class of #249259
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

812 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 812 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 128

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

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

Polycyclic presentation:

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

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, 158582);