#234903

order	512
nilpotency class	3
isoclinism	isoclinic class of #234892
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1296 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=i,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gi,c^b=cg,d^b=di,e^b=eh,d^c=di,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, 142099);