#10128232

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4069 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,d^2=ghi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10035428);