#116912

order	512
nilpotency class	3
isoclinism	isoclinic class of #116912
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

669 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 669 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,e^2=hi,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,e^b=ei,e^c=ehi,g^c=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 24108);