#9206620

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
prime radical	#55468	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3939 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,d^2=h,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi

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, 9113816);