#320761

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

Polycyclic presentation:

2^9|a^2=f,b^2=gh,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, 227957);