#101878

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

462 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 80 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)(3,5,7,8)(4,6)(10,14,17,20)(11,18,19,21)(25,26,29,33,30,28,37,32)(27,31,35,39,36,34,40,38);(1,3,4,7)(2,5)(6,8)(9,10,15,14,13,17,12,20)(11,16,18,22,19,24,21,23)(25,27,29,35,30,36,37,40)(26,31,33,39,28,34,32,38);(9,11,13,19)(10,16,17,24)(12,21,15,18)(14,22,20,23)(25,28)(26,30)(27,34)(29,32)(31,36)(33,37)(35,38)(39,40)

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=gh,d^2=gh,e^2=ghi,f^2=i,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, 9074);