#108689

order	512
nilpotency class	3
isoclinism	isoclinic class of #108682
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1434
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

455 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 455 conjugacy classes of subgroups — too many to draw. Only its 148 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)(3,10,12,6)(4,9)(5,8)(7,11,14,16)(13,15)(17,18,21,26,23,25,22,20)(19,27,28,24,30,32,29,31);(1,3)(2,6)(4,11)(5,12)(7,9)(8,10)(13,16)(14,15)(17,19,22,29,23,30,21,28)(18,24,26,32,25,31,20,27);(1,3)(2,7,8,14)(4,11)(5,12)(6,9,10,15)(13,16)(17,20,21,25,23,26,22,18)(19,27,28,31,30,32,29,24)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,d^2=i,e^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gi,f^b=fhi,e^c=ei,f^c=fh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15885);