#292069

order	512
nilpotency class	3
isoclinism	isoclinic class of #272795
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

802 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)(9,10,14,18)(11,15,20,23)(12,21,19,24)(13,17,22,16)(25,26)(27,34)(28,32)(29,39)(30,33)(31,37)(35,40)(36,38);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,22)(13,19)(14,20)(16,21)(17,24)(18,23)(25,27,30,37)(26,31,33,34)(28,38,39,35)(29,36,32,40);(9,12,13,11,14,19,22,20)(10,15,16,21,18,23,17,24)(25,28,30,39)(26,32,33,29)(27,35)(31,40)(34,36)(37,38);(2,6)(5,7)(25,29,30,32)(26,28,33,39)(27,36,37,40)(31,38,34,35)

Polycyclic presentation:

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

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, 199265);