#273674

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

1085 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)(11,18,13,16)(12,23)(14,19)(15,20)(17,26)(21,34,25,31)(22,36,28,32)(24,40,39,33)(27,38)(29,37)(30,35);(2,6)(3,4)(7,8)(9,11)(10,16)(12,24)(13,14)(15,28)(17,33)(18,19)(20,36)(21,27)(22,29)(23,40)(25,30)(26,39)(31,35)(32,37)(34,38);(9,12,14,26)(10,17,19,23)(11,21,13,25)(15,30,29,27)(16,31,18,34)(20,38,37,35)(22,39,28,24)(32,40,36,33);(1,3)(4,5)(6,7)(9,13,14,11)(10,18,19,16)(12,25,26,21)(15,28,29,22)(17,34,23,31)(20,36,37,32)(24,27,39,30)(33,35,40,38)

Polycyclic presentation:

2^9|a^2=fhi,c^2=hi,d^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, 180870);