#101828

order	512
nilpotency class	3
isoclinism	isoclinic class of #101799
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#386
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

356 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 86 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=g,d^2=gh,e^2=ghi,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,e^b=egh,f^b=fh,d^c=dgh,e^c=eghi,f^c=fgh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9024);