#9016442

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	#44258	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4007 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=i,d^2=ghi,e^2=i,f^2=h,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, 8923638);