#102922

order	512
nilpotency class	3
isoclinism	isoclinic class of #102905
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	#120
commutator, Frattini, cosocle	C₂²×C₄²	C₂³
LCS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

689 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 689 conjugacy classes of subgroups — too many to draw. Only its 140 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) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 36 points:

(6,9)(7,11)(10,12)(14,17)(15,19)(18,20)(21,22,25,28)(23,26,30,33)(24,27,31,34)(29,35,36,32);(1,2)(3,4)(5,6,8,10)(7,12,11,9)(21,23)(22,26)(24,29)(25,30)(27,32)(28,33)(31,36)(34,35);(2,4)(13,14,16,18)(15,20,19,17)(21,24)(22,27)(23,29)(25,31)(26,32)(28,34)(30,36)(33,35)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,d^2=g,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,c^b=cf,d^b=dg,e^b=eh,d^c=dh,e^c=ei

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, 10118);