#251328

order	512
nilpotency class	3
isoclinism	isoclinic class of #249019
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

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

normal subgroups only

This group has 906 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)(11,14)(12,15)(13,16)(17,21)(18,20)(19,22)(23,24)(25,26,29,33)(27,35,28,31)(30,39,38,34)(32,36,40,37);(1,3,4,8)(2,5,6,7)(9,11)(10,14)(12,17)(13,18)(15,21)(16,20)(19,23)(22,24)(25,27,29,28)(26,31,33,35)(30,37,38,36)(32,39,40,34);(14,20)(21,24)(25,28,29,27)(26,32,33,40)(30,36,38,37)(31,34,35,39);(2,6)(5,7)(9,12,13,19)(10,15,16,22)(11,17,18,23)(14,21,20,24)(26,33)(27,28)(34,39)(36,37)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=h,d^2=hi,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=dh,f^c=fh

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, 158524);