#331899

order	512
nilpotency class	3
isoclinism	isoclinic class of #331791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

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

normal subgroups only

This group has 620 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) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 239095);