#264025

order	512
nilpotency class	3
isoclinism	isoclinic class of #264023
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

872 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)(3,6)(4,8)(5,7)(9,10,16,22)(11,23,26,40)(12,27,30,35)(13,31,33,39)(14,21,15,20)(17,28,36,25)(18,34,38,29)(19,32,37,24);(1,2,4,8)(3,7,5,6)(9,11,14,25,16,26,15,28)(10,17,20,23,22,36,21,40)(12,24,29,33,30,32,34,13)(18,37,27,39,38,19,35,31);(2,6)(3,5)(7,8)(9,12)(10,18)(11,24)(13,28)(14,29)(15,34)(16,30)(17,19)(20,27)(21,35)(22,38)(23,31)(25,33)(26,32)(36,37)(39,40);(1,3,4,5)(2,7,8,6)(9,13,15,32,16,33,14,24)(10,19,21,39,22,37,20,31)(11,12,28,34,26,30,25,29)(17,35,40,38,36,27,23,18)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 171221);