#251439

order	512
nilpotency class	3
isoclinism	isoclinic class of #246751
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	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

790 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 790 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 158635);