#148585

order	512
nilpotency class	4
isoclinism	isoclinic class of #148585
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	C₂⁵	C₂²×C₄
Frattini, prime radical, cosocle	C₂³×D₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ C₂×#518 ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 717 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)(3,11)(4,15)(5,19,20,9)(6,10)(7,13)(8,17,24,32)(12,26)(14,22,28,25)(16,30)(18,29)(21,31,23,27);(1,3)(2,7,10,11)(4,16)(5,14)(6,13)(8,23,24,21)(9,22,19,25)(12,18)(15,26,29,30)(17,31)(20,28)(27,32);(1,4)(2,8)(3,12,13,16)(5,17)(6,18)(7,21,11,23)(9,15)(10,24)(14,27,28,31)(19,29)(20,32)(22,30,25,26)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,b^a=bd,c^a=ce,e^a=eh,h^a=hi,c^b=cf,d^b=di,f^c=fi,g^c=gh,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, 55781);