#9712432

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

4096 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)(3,6)(4,11)(5,7)(8,12)(9,15)(10,13)(14,16)(17,18)(19,21)(20,29)(22,31)(23,24)(25,30)(26,28)(27,32)(33,34,39,43)(35,44,46,40)(36,41,47,37)(38,45,48,42);(1,3,5,10)(2,6,7,13)(4,9,12,16)(8,14,11,15)(17,19,23,27)(18,21,24,32)(20,28,30,22)(25,31,29,26)(33,35,39,46)(34,40,43,44)(36,45,47,42)(37,48,41,38);(2,7)(6,13)(8,11)(14,15)(18,24)(19,27)(20,30)(26,31)(33,36,39,47)(34,37,43,41)(35,45,46,42)(38,40,48,44);(1,2)(3,6)(4,8)(5,7)(9,14)(10,13)(11,12)(15,16)(17,20)(18,25)(19,28)(21,31)(22,27)(23,30)(24,29)(26,32)(33,36)(34,41)(35,45)(37,43)(38,40)(39,47)(42,46)(44,48);(1,3)(2,6)(4,9)(5,10)(7,13)(8,14)(11,15)(12,16)(17,21)(18,19)(20,31)(22,29)(23,32)(24,27)(25,28)(26,30)(33,37,39,41)(34,36,43,47)(35,38,46,48)(40,42,44,45);(1,4,5,12)(2,8,7,11)(3,9,10,16)(6,14,13,15)(17,22,23,28)(18,26,24,31)(19,20,27,30)(21,25,32,29)(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=g,e^2=g,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, 9619628);