#276727

order	512
nilpotency class	3
isoclinism	isoclinic class of #268763
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

524 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=i,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,g^b=gi,f^c=fh,g^c=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 183923);