#273671

order	512
nilpotency class	3
isoclinism	isoclinic class of #273563
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1056 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)(3,7)(4,6)(5,8)(9,10)(11,19,23,16)(12,14)(13,24,28,17)(15,18)(20,22,34,30)(21,31)(25,29)(26,32)(27,37,40,33)(35,36)(38,39);(2,6)(3,4)(7,8)(9,11)(10,16)(12,19)(13,26)(14,23)(15,30)(17,32)(18,34)(20,25)(21,28)(22,29)(24,31)(27,36)(33,35)(37,38)(39,40);(9,12,14,10)(11,20,23,34)(13,24,28,17)(15,18,29,25)(16,30,19,22)(21,35,26,38)(27,33,40,37)(31,39,32,36);(1,3)(4,5)(6,7)(9,13)(10,17)(11,21)(12,24)(14,28)(15,27)(16,31)(18,33)(19,32)(20,35)(22,36)(23,26)(25,37)(29,40)(30,39)(34,38)

Polycyclic presentation:

2^9|a^2=fhi,c^2=hi,e^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,e^b=ei,f^b=fhi,g^b=ghi,e^c=ehi,g^c=ghi,e^d=ei,f^d=fhi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 180867);