#273717

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
prime radical	#62090	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

852 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=hi,c^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, 180913);