#232638

order	512
nilpotency class	3
isoclinism	isoclinic class of #232619
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×#121	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1162 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,6,10)(3,11,13,15)(4,9,5,7)(8,14,16,12)(17,18,23,28)(19,29,33,40)(20,34,38,43)(21,27,22,26)(24,32,41,31)(25,36,44,37)(30,39,46,47)(35,48,45,42);(2,7)(3,12)(4,5)(9,10)(11,15)(13,14)(17,19,23,33)(18,24,28,41)(20,35,38,45)(21,32,22,31)(25,42,44,48)(26,40,27,29)(30,37,46,36)(34,47,43,39);(1,3)(2,8)(4,12)(5,14)(6,13)(7,11)(9,15)(10,16)(17,20)(18,25)(19,30)(21,36)(22,37)(23,38)(24,39)(26,43)(27,34)(28,44)(29,42)(31,35)(32,45)(33,46)(40,48)(41,47);(1,4,6,5)(2,9,10,7)(3,12,13,14)(8,15,16,11)

Polycyclic presentation:

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

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, 139834);