#93238

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1351
commutator	C₂×#95	C₂×C₄
Frattini, cosocle	#1446	C₂²
prime radical	C₂⁴	#99
LCS	G ▷ C₂×#95 ▷ C₂³×C₄ ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×C₄ ◁ #1446 ◁ G
LECS	G ▷ #1446 ▷ C₂³×C₄ ▷ C₂³ ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 228 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fg,d^2=h,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, 434);