#9719169

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	#45649	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3999 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)(3,8)(4,13)(5,10)(6,14)(7,9)(11,16)(12,15)(17,18)(19,29)(20,23)(21,26)(22,27)(24,28)(25,32)(30,31)(33,34,39,43)(35,42,38,40)(36,37,41,48)(44,45,46,47);(1,3,7,11)(2,8,9,16)(4,5,14,15)(6,12,13,10)(17,19)(18,25)(20,22)(21,30)(23,31)(24,32)(26,27)(28,29)(33,35,39,38)(34,40,43,42)(36,44,41,46)(37,47,48,45);(2,9)(3,11)(4,14)(10,12)(17,20,24,21)(18,23,28,26)(19,22,32,30)(25,31,29,27)(33,35,39,38)(34,40,43,42)(36,46,41,44)(37,45,48,47);(1,4)(2,6)(3,5)(7,14)(8,12)(9,13)(10,16)(11,15)(17,21,24,20)(18,26,28,23)(19,30,32,22)(25,27,29,31)(33,36,39,41)(34,37,43,48)(35,44,38,46)(40,47,42,45);(1,5)(2,10)(3,4)(6,16)(7,15)(8,13)(9,12)(11,14)(17,22,24,30)(18,27,28,31)(19,20,32,21)(23,25,26,29)(33,37)(34,41)(35,45)(36,43)(38,47)(39,48)(40,44)(42,46);(1,6)(2,4)(3,12)(5,8)(7,13)(9,14)(10,11)(15,16)(17,23,24,26)(18,20,28,21)(19,31,32,27)(22,29,30,25)(33,38,39,35)(34,42,43,40)(36,46,41,44)(37,45,48,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,d^2=g,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9626365);