#292724

order	512
nilpotency class	3
isoclinism	isoclinic class of #271387
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	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

755 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)(4,6)(5,8)(9,10)(11,19)(12,18)(13,15)(14,17)(16,20)(21,24)(22,23)(25,26,30,34)(27,35,37,32)(28,31,33,29)(36,40,39,38);(1,3,4,8)(2,5,6,7)(10,15)(16,23)(17,18)(19,24)(25,27,30,37)(26,32,34,35)(28,38,33,40)(29,36,31,39);(9,11,14,20,13,21,12,22)(10,16,18,19,15,23,17,24)(25,28,31,34,30,33,29,26)(27,35,36,38,37,32,39,40);(2,6)(5,7)(9,12,13,14)(10,17,15,18)(11,20,21,22)(16,19,23,24)(25,29,30,31)(26,33,34,28)(27,36,37,39)(32,40,35,38)

Polycyclic presentation:

2^9|a^2=i,b^2=ehi,c^2=f,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=dhi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 199920);