#232785

order	512
nilpotency class	3
isoclinism	isoclinic class of #232783
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#392
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1305 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)(3,8)(4,7)(5,9)(6,10)(11,14)(12,15)(13,16)(17,18)(19,28)(20,33)(21,38)(22,26)(23,27)(24,31)(25,36)(29,46)(30,41)(32,40)(34,45)(35,44)(37,43)(39,48)(42,47);(2,7)(3,11)(4,5)(9,10)(12,13)(14,15)(17,19)(18,24)(20,34)(21,30)(22,31)(23,32)(25,42)(26,40)(27,41)(28,38)(29,35)(33,39)(36,48)(37,47)(43,46)(44,45);(1,3)(2,8)(4,11)(5,12)(6,13)(7,14)(9,15)(10,16)(17,20)(18,25)(19,29)(21,35)(22,36)(23,37)(24,39)(26,43)(27,44)(28,45)(30,47)(31,34)(32,48)(33,38)(40,42)(41,46);(1,4,6,5)(2,9,10,7)(3,11,13,12)(8,15,16,14)(17,21,23,22)(18,26,27,38)(19,30,32,31)(20,35,37,36)(24,40,41,28)(25,43,44,33)(29,47,48,34)(39,42,46,45)

Polycyclic presentation:

2^9|d^2=h,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fhi,g^a=gi,c^b=cg,d^b=di,e^b=eh,g^b=gh,e^c=ei,f^c=fhi,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, 139981);