#120229

order	512
nilpotency class	3
isoclinism	isoclinic class of #120226
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1441
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#117
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

460 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 134 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)(3,11)(4,7)(5,14)(6,10)(8,13)(9,15)(12,16)(17,18,21,24,20,23,28,31)(19,25,27,32,26,22,29,30);(1,3,5,4,6,12,15,13)(2,7,9,11,10,8,14,16)(17,19,20,26)(18,22,23,25)(21,27,28,29)(24,30,31,32);(1,4,5,12,6,13,15,3)(2,8,9,16,10,7,14,11)(18,23)(21,28)(27,29)(30,32)

Polycyclic presentation:

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

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, 27425);