#94528

order	512
nilpotency class	6
isoclinism	isoclinic class of #94526
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7339
commutator	C₂×C₄²	C₂×C₈
Frattini, cosocle	#1822	C₂²
prime radical	C₂×#540	C₄
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ #1822 ◁ G
LECS	G ▷ #1822 ▷ C₂³×C₄ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 230 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:

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

Polycyclic presentation:

2^9|a^2=d,c^2=h,d^2=f,e^2=i,f^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, 1724);