#94526

order	512
nilpotency class	6
isoclinism	isoclinic class of #94526
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7339
commutator	C₂×C₄²	C₂×C₈
Frattini, cosocle	#1820	C₂²
prime radical	C₂×#540	C₄
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂⁴ ◁ C₂³×C₄ ◁ #1820 ◁ G
LECS	G ▷ #1820 ▷ C₂³×C₄ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁

234 conjugacy classes of subgroups; only the 24 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 24 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:

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

Polycyclic presentation:

2^9|a^2=d,c^2=h,d^2=f,e^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=ch,d^b=dehi,e^b=ei,f^b=fhi,d^c=dgi,f^c=fi,e^d=eh,g^d=gi

Conjugacy classes & character table

29 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1722);