#9232833

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	#44065	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4078 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,d^2=i,e^2=gh,f^2=gi,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, 9140029);