#8944433

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

4138 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 8851629);