#8912223

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

4195 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)(3,7,10,4)(5,8,14,16)(11,13,15,12)(18,23)(19,26)(25,31)(27,28)(33,34,38,43)(35,39,45,47)(36,41,46,40)(37,44,48,42);(1,3)(2,7)(4,9)(5,11)(6,10)(8,13)(12,16)(14,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,32)(28,29)(30,31)(33,35,38,45)(34,39,43,47)(36,42,46,44)(37,40,48,41);(3,10)(4,7)(11,15)(12,13)(19,26)(21,24)(27,28)(29,32)(33,36,38,46)(34,40,43,41)(35,44,45,42)(37,47,48,39);(1,4,6,7)(2,3,9,10)(5,13,14,12)(8,15,16,11)(17,19,22,26)(18,24,23,21)(20,28,30,27)(25,29,31,32)(33,36)(34,41)(35,42)(37,39)(38,46)(40,43)(44,45)(47,48);(1,5)(2,8)(3,11)(4,12)(6,14)(7,13)(9,16)(10,15)(17,20,22,30)(18,25,23,31)(19,27,26,28)(21,29,24,32)(33,35)(34,39)(36,44)(37,40)(38,45)(41,48)(42,46)(43,47);(1,3)(2,7)(4,9)(5,11)(6,10)(8,13)(12,16)(14,15)(17,21,22,24)(18,26,23,19)(20,29,30,32)(25,28,31,27)(33,37,38,48)(34,42,43,44)(35,40,45,41)(36,39,46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=ghi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8819419);