#9127572

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4052 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=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, 9034768);