#9026769

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	#48174	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3911 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=gi,d^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8933965);