#234937

order	512
nilpotency class	3
isoclinism	isoclinic class of #234934
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60359	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2239 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|c^2=h,d^2=h,e^2=h,b^a=be,c^a=cf,e^a=eh,g^a=ghi,c^b=cg,e^b=eh,f^b=fh,d^c=dh,e^c=ei

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, 142133);