#8605686

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4178 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=hi,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=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, 8512882);