#254853

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	#46133	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

879 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)(11,14)(12,20)(13,19)(15,17)(16,24)(18,22)(21,23)(25,26,31,35)(27,32,28,37)(29,34,30,39)(33,36,40,38);(2,5)(6,8)(9,11,14,10)(12,21,23,20)(13,19,15,17)(16,22,18,24)(25,27,31,28)(26,32,35,37)(29,36,30,38)(33,34,40,39);(1,3,4,7)(2,6,5,8)(9,12,14,23)(10,16,11,18)(13,22,15,24)(17,21,19,20)(25,28,31,27)(26,33,35,40)(29,38,30,36)(32,39,37,34);(2,5)(6,8)(9,13,14,15)(10,17,11,19)(12,22,23,24)(16,21,18,20)(25,29,31,30)(26,34,35,39)(27,36,28,38)(32,40,37,33)

Polycyclic presentation:

2^9|a^2=i,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, 162049);