#122813

order	512
nilpotency class	4
isoclinism	isoclinic class of #122724
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1519
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₂²⋊C₈	C₂³
prime radical	C₂²×D₄	SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

314 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 68 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)(3,9,10,16)(4,12,14,8)(5,7)(6,11)(13,15)(17,18,21,27,23,29,22,28)(19,26,30,24,31,32,20,25);(1,3,5,10)(2,6,7,15)(4,13,14,11)(8,16,12,9)(18,24)(19,30)(20,31)(21,22)(25,29)(26,27)(28,32);(2,7)(3,10)(4,14)(6,15)(17,19,22,20,23,31,21,30)(18,25,27,26,29,24,28,32)

Polycyclic presentation:

2^9|a^2=e,b^2=hi,c^2=ghi,d^2=gi,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gi,c^b=cfhi,d^b=dg,e^b=efgi,f^b=fi,g^b=gi,d^c=dhi,e^d=ehi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30009);