#273503

order	512
nilpotency class	3
isoclinism	isoclinic class of #273435
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#61218	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

950 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=fi,c^2=h,e^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,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

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 180699);