#101889

order	512
nilpotency class	3
isoclinism	isoclinic class of #101799
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
prime radical	C₂×#127	C₂×C₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

444 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 78 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 — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,d^2=gh,e^2=ghi,f^2=gi,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, 9085);