#102368

order	512
nilpotency class	3
isoclinism	isoclinic class of #102147
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#390
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

340 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 102 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 — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=egh,d^2=gh,e^2=i,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,f^b=fh

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9564);