#303920

order	512
nilpotency class	3
isoclinism	isoclinic class of #269211
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2819	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

790 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)(9,10,13,17)(11,19,20,16)(12,15,18,14)(21,24,23,22)(25,26,30,33)(27,34,36,31)(28,37,38,35)(29,39,40,32);(1,3,5,4)(2,6,8,7)(9,11)(10,16)(12,22)(13,20)(14,21)(15,23)(17,19)(18,24)(25,27)(26,31)(28,32)(29,35)(30,36)(33,34)(37,40)(38,39);(1,3,5,4)(2,7,8,6)(9,12,14,10,13,18,15,17)(11,16,21,24,20,19,23,22)(25,28,29,31,30,38,40,34)(26,27,32,37,33,36,39,35);(1,4,5,3)(2,7,8,6)(10,17)(12,18)(16,19)(22,24)(25,29,30,40)(26,32,33,39)(27,35,36,37)(28,34,38,31)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=efh,d^2=ehi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,d^c=dh,g^c=gh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 211116);