#93243

order	512
nilpotency class	5
isoclinism	isoclinic class of #93229
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#353
commutator	C₂×#97	C₂×C₄
Frattini, cosocle	#1985	C₂²
socle	C₂²	#1352
prime radical	C₂³	#328
LCS	G ▷ C₂×#97 ▷ C₂×C₄² ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1985 ◁ G
LECS	G ▷ #1985 ▷ C₂×C₄² ▷ C₂³ ▷ C₂² ▷ C₁

135 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 135 conjugacy classes of subgroups — too many to draw. Only its 26 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=eg,c^2=fghi,d^2=h,e^2=hi,f^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,c^b=cf,d^b=defgi,e^b=ei,g^b=gi,d^c=dghi,e^c=ei,e^d=eh

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 439);