#9029106

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

4144 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gi,d^2=h,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8936302);