#135336

order	512
nilpotency class	4
isoclinism	isoclinic class of #124657
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#99
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁵ ◁ C₂³×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

726 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 100 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,26,21,25,20,24)(19,27,28,32,29,23,30,31);(2,5)(6,8)(10,13)(14,15)(17,19)(18,23)(20,30)(21,29)(22,28)(24,32)(25,27)(26,31);(1,3,4,7)(2,6,5,8)(10,14)(12,16)(13,15)(17,20,21,22)(18,24)(19,28,29,30)(23,31)(25,26)(27,32)

Polycyclic presentation:

2^9|a^2=f,c^2=dg,e^2=hi,f^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gi,c^b=ci,e^c=eh,f^c=fghi,f^e=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42532);