#120516

order	512
nilpotency class	3
isoclinism	isoclinic class of #120516
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1441
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

706 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 706 conjugacy classes of subgroups — too many to draw. Only its 132 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)(4,6)(7,8)(9,10)(11,19)(12,16,24,30)(13,25)(14,18)(15,21)(17,26)(20,31,34,40)(22,32)(23,36,37,29)(27,38,39,33)(28,35);(1,3)(2,5)(4,7)(6,8)(9,11,13,21,14,22,26,35)(10,15,17,19,18,28,25,32)(12,20,23,33,24,34,37,38)(16,27,29,31,30,39,36,40);(5,8)(9,12,13,23,14,24,26,37)(10,16,17,29,18,30,25,36)(11,20,21,33,22,34,35,38)(15,27,19,31,28,39,32,40)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=d,d^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,e^b=ei,e^c=eh,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27712);