#8917890

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

4221 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)(3,9)(4,12)(5,11)(6,7)(8,13)(10,15)(14,16)(33,34,39,38)(35,44,46,40)(36,41,45,42)(37,43,48,47);(1,3,6,11)(2,5,7,9)(4,10,13,14)(8,15,12,16)(17,18,22,23)(19,24,26,31)(20,21,25,30)(27,32,29,28)(33,35,39,46)(34,40,38,44)(36,37,45,48)(41,47,42,43);(2,7)(4,13)(5,9)(10,14)(18,23)(19,26)(21,30)(27,29)(33,36,39,45)(34,41,38,42)(35,37,46,48)(40,47,44,43);(1,4)(2,8)(3,10)(5,15)(6,13)(7,12)(9,16)(11,14)(17,19)(18,24)(20,29)(21,28)(22,26)(23,31)(25,27)(30,32)(33,36)(34,42)(35,37)(38,41)(39,45)(40,43)(44,47)(46,48);(1,3)(2,9)(4,14)(5,7)(6,11)(8,15)(10,13)(12,16)(17,20,22,25)(18,21,23,30)(19,27,26,29)(24,32,31,28)(33,37)(34,43)(35,45)(36,46)(38,47)(39,48)(40,41)(42,44);(1,5,6,9)(2,3,7,11)(4,15,13,16)(8,10,12,14)(17,21)(18,25)(19,28)(20,23)(22,30)(24,27)(26,32)(29,31)(33,38,39,34)(35,44,46,40)(36,41,45,42)(37,47,48,43)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8825086);