#144112

order	512
nilpotency class	4
isoclinism	isoclinic class of #144077
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2011
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#447
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

412 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 84 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)(3,11)(4,8)(5,14)(6,10)(7,12)(9,15)(13,16)(17,18,21,25)(19,22,28,31)(20,29,27,24)(23,32,30,26);(1,3,6,4)(2,7,10,16)(5,13,15,12)(8,14,11,9)(18,22)(19,26)(20,27)(23,24)(25,31)(28,32)(29,30);(1,4,6,3)(2,8,10,11)(5,12,15,13)(7,9,16,14)(18,22)(19,26)(20,27)(23,24)(25,31)(28,32)(29,30)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51308);