#144244

order	512
nilpotency class	4
isoclinism	isoclinic class of #144210
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	#33752	C₂
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

511 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 511 conjugacy classes of subgroups — too many to draw. Only its 76 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)(3,6)(4,7)(5,8)(9,14)(10,13)(11,15)(12,16)(17,18)(19,22)(20,29)(21,25)(23,26)(24,27)(28,31)(30,32);(1,3)(2,6)(4,9)(5,10)(7,14)(8,13)(11,16)(12,15)(18,22)(19,26)(20,27)(23,24)(25,31)(28,32)(29,30);(1,4,5,11)(2,7,8,15)(3,9,10,16)(6,12,13,14)

Polycyclic presentation:

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

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, 51440);