#115316

order	512
nilpotency class	3
isoclinism	isoclinic class of #115302
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1036 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 1036 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,c^b=cf,d^b=dh,d^c=di,g^c=ghi

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, 22512);