#289112

order	512
nilpotency class	3
isoclinism	isoclinic class of #289075
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

630 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=ef,c^2=i,d^2=ehi,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,g^b=gh,f^c=fi,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 196308);