#9712139

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

4099 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)(3,7)(4,12)(5,6)(8,14)(9,11)(10,16)(13,15)(17,18,23,21)(19,25,20,24)(22,26,30,29)(27,32,28,31)(33,34)(35,38)(36,43)(37,40)(39,44)(41,47)(42,46)(45,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,16)(12,14)(13,15)(17,19,23,20)(18,24,21,25)(22,27,30,28)(26,31,29,32)(33,35,37,42)(34,38,40,46)(36,41,44,48)(39,45,43,47);(2,6)(7,9)(8,12)(15,16)(33,34,37,40)(35,38,42,46)(36,39,44,43)(41,45,48,47);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,14)(13,16)(17,20,23,19)(18,25,21,24)(22,28,30,27)(26,32,29,31)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,3)(2,7)(4,13)(5,11)(6,9)(8,16)(10,14)(12,15)(17,21,23,18)(19,25,20,24)(22,26,30,29)(27,31,28,32)(33,35)(34,38)(36,41)(37,42)(39,45)(40,46)(43,47)(44,48);(1,4)(2,8)(3,10)(5,14)(6,12)(7,15)(9,16)(11,13)(17,22)(18,26)(19,27)(20,28)(21,29)(23,30)(24,31)(25,32)(33,36,37,44)(34,39,40,43)(35,41,42,48)(38,45,46,47)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=hi,d^2=g,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,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, 9619335);