#94622

order	512
nilpotency class	6
isoclinism	isoclinic class of #94622
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7394
commutator	#381	C₂×C₄
Frattini, cosocle	#1782	C₂²
prime radical	#9980	C₂
LCS	G ▷ #381 ▷ C₂³×C₄ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂⁴ ◁ C₂³×C₄ ◁ #1782 ◁ G
LECS	G ▷ #1782 ▷ C₂³×C₄ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁

266 conjugacy classes of subgroups; only the 22 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 22 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)(3,12,13,25)(5,10,16,26)(6,11,17,27)(7,20,21,19)(8,23,24,14)(15,22,29,32)(18,28,30,31);(2,7)(4,16)(5,18)(8,11)(9,25)(10,12)(13,23)(14,19)(17,30)(20,29)(21,31)(22,28)(24,26)(27,32)

Polycyclic presentation:

2^9|a^2=d,c^2=f,e^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fh,h^a=hi,c^b=cf,d^b=defi,e^b=ei,g^b=gh,d^c=dgi,e^c=ehi,g^c=gi,f^d=fi

Conjugacy classes & character table

26 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1818);