#143096

order	512
nilpotency class	4
isoclinism	isoclinic class of #143094
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#28149	C₂
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#436 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

441 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 441 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50292);