#8841638

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

4250 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8748834);