#121967

order	512
nilpotency class	3
isoclinism	isoclinic class of #99134
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#381
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

550 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=deh,d^2=hi,e^2=hi,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,g^b=gi,g^c=gh

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, 29163);