#148586

order	512
nilpotency class	4
isoclinism	isoclinic class of #148586
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11197
commutator	C₂³×C₄	C₂²×C₄
Frattini, prime radical, cosocle	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ C₂×#518 ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂² ▷ C₂ ▷ C₁

733 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 733 conjugacy classes of subgroups — too many to draw. Only its 120 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)(3,7,12,22)(4,14,17,30)(5,18,19,10)(6,9)(8,16)(11,25)(13,28)(15,29)(20,31,23,26)(21,27)(24,32);(1,3)(2,7,9,22)(4,15)(5,13)(6,12)(8,23,24,20)(10,21,18,28)(11,17)(14,29,30,25)(16,31)(19,27)(26,32);(1,4)(2,8,9,24)(3,11,12,15)(5,16)(6,17)(7,20)(10,14,18,30)(13,26,27,31)(19,32)(21,25)(22,23)(28,29)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,e^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,h^a=hi,c^b=cf,d^b=di,g^b=gi,f^c=fi,g^c=ghi,g^e=gi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55782);