#9066068

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	#54823	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4045 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8973264);