#276378

order	512
nilpotency class	3
isoclinism	isoclinic class of #268891
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

581 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)(3,7)(4,5)(6,8)(9,10,15,23,14,19,13,18)(11,16,20,24,12,22,21,17)(25,26,28,33,29,32,38,31)(27,34,35,40,37,30,36,39);(2,5)(6,7)(9,11)(10,16)(12,14)(13,21)(15,20)(17,18)(19,22)(23,24)(25,27,29,37)(26,30,32,34)(28,35,38,36)(31,40,33,39);(9,12,14,11)(10,17,19,24)(13,21,15,20)(16,23,22,18)(26,31)(27,35)(28,38)(32,33)(36,37)(39,40);(1,3,4,8)(2,6,5,7)(9,13,14,15)(10,18,19,23)(11,20,12,21)(16,24,22,17)(25,28,29,38)(26,31,32,33)(27,36,37,35)(30,39,34,40)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 183574);