#9236457

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	#2631	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3818 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,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, 9143653);