#320809

order	512
nilpotency class	3
isoclinism	isoclinic class of #298939
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

648 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=fh,b^2=g,c^2=hi,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,f^c=fh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 228005);