#135527

order	512
nilpotency class	4
isoclinism	isoclinic class of #135527
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8747
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 529 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)(3,11,15,28)(4,16,12,7)(5,19)(6,10)(8,14,25,21)(9,20)(13,30)(17,27)(18,23,29,26)(22,32)(24,31);(2,7)(5,20)(8,22)(9,26)(10,16)(11,27)(14,21)(18,29)(19,23)(24,25)(28,30)(31,32);(1,3,12,13)(2,8,16,24)(4,17,6,15)(5,21,29,32)(7,22,10,25)(9,11,26,27)(14,18,31,20)(19,28,23,30)

Polycyclic presentation:

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

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42723);