#94532

order	512
nilpotency class	6
isoclinism	isoclinic class of #94532
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7344
commutator	#127	C₂×C₈
Frattini, cosocle	#1961	C₂²
prime radical	#3543	C₄
LCS	G ▷ #127 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1961 ◁ G
LECS	G ▷ #1961 ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 241 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) = 32

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

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

Polycyclic presentation:

2^9|a^2=d,d^2=f,e^2=h,g^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi,d^b=deh,e^b=eh,f^b=fi,g^b=gh,d^c=dghi,e^c=eh,g^c=gi,e^d=ehi,g^d=gi

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1728);