#324607

order	512
nilpotency class	3
isoclinism	isoclinic class of #278987
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

546 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 150 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)(4,5)(6,8)(9,10,12,15)(11,17,20,14)(13,16,18,22)(19,24,21,23)(25,26)(27,34)(28,30)(29,32)(31,36)(33,35)(37,40)(38,39);(2,5)(6,7)(9,11,18,19,12,20,13,21)(10,14,22,23,15,17,16,24)(25,27,35,39,29,36,30,37)(26,31,33,40,32,34,28,38);(11,19)(13,18)(14,23)(16,22)(17,24)(20,21)(25,28,35,32,29,33,30,26)(27,31,39,38,36,34,37,40);(1,3,4,8)(2,6,5,7)(27,36)(31,34)(37,39)(38,40)

Polycyclic presentation:

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

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, 231803);