#130135

order	512
nilpotency class	4
isoclinism	isoclinic class of #130074
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1790
commutator	C₂×C₄⋊C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 549 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,d^2=h,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,c^b=ci,d^b=dh,e^b=egi,f^b=fgh,d^c=dg,e^d=ei,f^d=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, 37331);