#9236464

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	#50715	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4106 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gi,d^2=h,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,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, 9143660);