#93237

order	512
nilpotency class	5
isoclinism	isoclinic class of #93237
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1351
commutator	C₂×#95	C₂×C₄
Frattini, cosocle	#1438	C₂²
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂×#95 ▷ C₂³×C₄ ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×C₄ ◁ #1438 ◁ G
LECS	G ▷ #1438 ▷ C₂³×C₄ ▷ C₂³ ▷ C₂² ▷ C₁

297 conjugacy classes of subgroups; only the 28 normal subgroups are shown.

normal subgroups only

This group has 297 conjugacy classes of subgroups — too many to draw. Only its 28 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) = 32

A minimal faithful permutation representation — 2 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fg,e^2=h,b^a=bc,c^a=ce,e^a=eg,f^a=fi,g^a=gh,c^b=cf,d^b=defgh,f^b=fi,g^b=gi,d^c=dgh,e^c=ei,e^d=eh

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 433);