#260045

order	512
nilpotency class	3
isoclinism	isoclinic class of #246239
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	C₂³×C₄	C₂⁴
prime radical	#29712	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

998 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)(4,5)(6,8)(9,10)(11,19)(12,17)(13,16)(14,18)(15,21)(20,24)(22,23)(25,26)(27,33)(28,31)(29,32)(30,35)(34,37)(36,39)(38,40);(2,5)(6,7)(9,11,12,21)(10,15,17,19)(13,22,14,20)(16,24,18,23)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,40)(37,39);(10,16)(11,20)(13,14)(17,18)(21,22)(23,24)(25,28)(26,31)(27,34)(29,36)(30,37)(32,39)(33,38)(35,40);(1,3,4,8)(2,6,5,7)(10,17)(15,19)(16,18)(23,24)(26,32)(28,36)(30,33)(34,40)

Polycyclic presentation:

2^9|b^2=h,d^2=ehi,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,f^c=fh,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, 167241);