#122084

order	512
nilpotency class	3
isoclinism	isoclinic class of #122083
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

650 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319662, #72319738
maximal normal subgroup of	#72319662, #72319738

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29280);