#144360

order	512
nilpotency class	4
isoclinism	isoclinic class of #144296
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2021
commutator, Frattini, cosocle	C₂×C₄×D₄	C₂³
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

552 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 58 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) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51556);