#331657

order	512
nilpotency class	3
isoclinism	isoclinic class of #331655
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#10228	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 748 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)(11,18)(12,14)(13,17)(15,19)(16,21)(20,24)(22,23)(25,26)(27,32)(28,36)(29,31)(30,33)(34,40)(35,38)(37,39);(1,3)(2,5)(4,8)(6,7)(26,31)(27,34)(28,30)(32,39)(35,37)(38,40);(10,14)(12,21)(15,18)(16,17)(19,22)(23,24)(25,27)(26,32)(28,34)(29,35)(30,37)(31,38)(33,39)(36,40);(2,6)(3,8)(9,11,13,20)(10,15,17,23)(12,22,21,19)(14,18,16,24)(25,28,29,30)(26,33,31,36)(27,34,35,37)(32,39,38,40)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 238853);