#135515

order	512
nilpotency class	4
isoclinism	isoclinic class of #135515
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8746
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

516 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 60 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) = 32

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=d,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,f^b=fg,e^c=egi,f^c=fhi,h^c=hi,f^d=fi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42711);