#329600

order	512
nilpotency class	3
isoclinism	isoclinic class of #329587
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

684 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 154 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)(3,6)(4,5)(7,8)(9,10,12,15,13,16,21,24)(11,17,18,22,19,23,20,14)(25,26,28,33,29,32,37,31)(27,30,34,39,36,38,35,40);(2,5)(6,8)(25,27,29,36)(26,30,32,38)(28,34,37,35)(31,40,33,39);(1,3)(2,6)(4,7)(5,8)(9,11,13,19)(10,14,16,22)(12,20,21,18)(15,23,24,17)(26,31)(27,34)(28,37)(32,33)(35,36)(39,40);(3,7)(6,8)(26,32)(30,38)(31,33)(39,40)

Polycyclic presentation:

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

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, 236796);