#8889142

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	#58064	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4028 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:

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

Polycyclic presentation:

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

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, 8796338);