#122000

order	512
nilpotency class	3
isoclinism	isoclinic class of #121999
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1446
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

650 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 650 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

(2,5,6,7)(9,10)(11,20)(12,24,27,39)(13,28)(14,18)(15,19)(16,22)(17,26,34,25)(21,30,37,33)(23,31)(29,32)(35,36,38,40);(1,2)(3,7)(4,6)(5,8)(9,11)(10,16,19,32)(12,21)(13,22)(14,29)(15,23)(17,33,34,30)(18,20,28,31)(24,38,39,35)(25,36)(26,40)(27,37);(9,12,15,27)(10,17)(11,21,23,37)(13,25,14,26)(16,30)(18,24)(19,34)(20,35)(22,36,29,40)(28,39)(31,38)(32,33)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=fh,d^2=hi,e^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=di,e^b=eh,g^b=gi,g^c=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224700, #72224701, #72224702
maximal normal subgroup of	#72224700, #72224701, #72224702

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29196);