#120539

order	512
nilpotency class	3
isoclinism	isoclinic class of #120533
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1441
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

453 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 453 conjugacy classes of subgroups — too many to draw. Only its 136 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27735);