#130091

order	512
nilpotency class	4
isoclinism	isoclinic class of #130089
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	C₂²×#41	D₄
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 630 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)(12,25,27,17)(13,18)(14,19)(15,22,32,21)(16,26,35,28)(23,30)(24,37)(29,34)(31,39,36,38)(33,40);(1,2)(3,7)(4,6)(5,8)(9,11,14,24)(10,15,19,32)(12,22,27,21)(13,23,29,40)(16,33,35,30)(17,20,25,37)(18,36,34,31)(26,39,28,38);(9,12,14,27)(10,16,19,35)(11,21,24,22)(13,28,29,26)(15,30,32,33)(17,34,25,18)(20,36,37,31)(23,39,40,38)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=h,e^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci,d^b=dh,e^b=eg,f^b=fgh,d^c=dgi,e^c=ei,f^c=fi,e^d=ei,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72228029, #72228034, #72228037
maximal normal subgroup of	#72228029, #72228034, #72228037

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 37287);