#232558

order	512
nilpotency class	3
isoclinism	isoclinic class of #232546
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

1230 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139754);