#234854

order	512
nilpotency class	3
isoclinism	isoclinic class of #234851
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	C₂³×#41	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2210 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)(9,10,15,20)(11,21,24,33)(12,25,26,34)(13,19,14,18)(16,28,31,23)(17,29,32,27)(22,30,36,40)(35,37,39,38);(5,8)(9,11)(10,16)(12,22)(13,28)(14,23)(15,24)(17,30)(18,33)(19,21)(20,31)(25,39)(26,36)(27,38)(29,37)(32,40)(34,35);(1,3)(2,5)(4,7)(6,8)(9,12,15,26)(10,17,20,32)(11,22,24,36)(13,27,14,29)(16,30,31,40)(18,25,19,34)(21,35,33,39)(23,37,28,38);(3,7)(5,8)(12,26)(17,32)(22,36)(25,34)(27,29)(30,40)(35,39)(37,38)

Polycyclic presentation:

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

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, 142050);