#135557

order	512
nilpotency class	4
isoclinism	isoclinic class of #135556
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8749
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

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

normal subgroups only

This group has 474 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)(3,12,16,26)(4,9,13,7)(5,19)(6,11)(8,15,23,31)(10,20)(14,28)(17,27)(18,22,29,25)(21,32)(24,30);(2,7,11,9)(5,20)(8,21,23,24)(10,25,19,22)(12,27,26,28)(15,31)(18,29)(30,32);(1,3,13,14,6,16,4,17)(2,8,7,21,11,23,9,24)(5,15,29,30,20,31,18,32)(10,26,22,27,19,12,25,28)

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=dh,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,e^b=ei,f^b=fgi,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, 42753);