#9221183

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	#2703	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3944 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=gi,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,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, 9128379);