#9206583

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	#49733	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3948 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=g,d^2=gh,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, 9113779);