#9203617

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	#2688	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3971 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)(3,7)(4,5)(6,9)(8,10)(11,12)(13,16)(14,15)(17,18,24,27)(19,26,23,21)(20,29,22,25)(28,32,31,30)(34,39)(35,41)(37,45)(44,46);(1,3)(2,7)(4,11)(5,12)(6,10)(8,9)(13,14)(15,16)(17,19,24,23)(18,21,27,26)(20,28,22,31)(25,32,29,30)(34,39)(36,42)(40,47)(44,46);(3,10)(7,8)(11,13)(12,16)(17,20,24,22)(18,25,27,29)(19,28,23,31)(21,32,26,30)(33,34,38,39)(35,42,41,36)(37,40,45,47)(43,46,48,44);(1,3)(2,8)(4,13)(5,12)(6,10)(7,9)(11,14)(15,16)(17,21,24,26)(18,19,27,23)(20,30,22,32)(25,31,29,28)(33,35,38,41)(34,36,39,42)(37,43,45,48)(40,44,47,46);(1,4,6,14)(2,5,9,15)(3,11,10,13)(7,12,8,16)(17,22,24,20)(18,25,27,29)(19,28,23,31)(21,30,26,32)(33,36)(34,35)(37,46)(38,42)(39,41)(40,48)(43,47)(44,45);(1,5)(2,4)(3,12)(6,15)(7,11)(8,13)(9,14)(10,16)(17,23,24,19)(18,26,27,21)(20,31,22,28)(25,30,29,32)(33,37,38,45)(34,40,39,47)(35,43,41,48)(36,44,42,46)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=h,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9110813);