#9726451

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4267 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=hi,d^2=gh,e^2=gi,f^2=gh,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, 9633647);