#273700

order	512
nilpotency class	3
isoclinism	isoclinic class of #273691
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
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

871 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,d^2=h,e^2=i,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,g^b=gh,e^c=ehi,f^c=fh,g^c=gh,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, 180896);