#140976

order	512
nilpotency class	4
isoclinism	isoclinic class of #140973
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1961
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	#33179	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

469 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 469 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=eg,d^2=ghi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,c^b=cg,d^b=dh,e^b=ei,f^b=fh,f^c=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48172);