#254854

order	512
nilpotency class	3
isoclinism	isoclinic class of #246751
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

830 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=ei,d^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,f^c=fh,g^c=ghi

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, 162050);