#291305

order	512
nilpotency class	3
isoclinism	isoclinic class of #269019
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	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

771 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 198501);