#94706

order	512
nilpotency class	6
isoclinism	isoclinic class of #94704
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7428
commutator	C₄³	C₂×C₄
Frattini, cosocle	#1962	C₂²
prime radical	#532	D₄
LCS	G ▷ C₄³ ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1962 ◁ G
LECS	G ▷ #1962 ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 141 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1902);