#9017460

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	#51125	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4095 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:

(1,2,6,10)(3,4,8,7)(5,13,15,9)(11,16,14,12)(17,18)(19,24)(20,23)(21,25)(22,26)(27,29)(28,31)(30,32);(1,3)(2,4)(5,11)(6,8)(7,10)(9,12)(13,16)(14,15)(17,19)(18,20)(21,27)(22,23)(24,26)(25,28)(29,32)(30,31)(33,34,39,43)(35,44,46,40)(36,38,42,48)(37,41,45,47);(1,3,6,8)(2,7,10,4)(5,14,15,11)(9,12,13,16)(17,20,22,24)(18,23,26,19)(21,29,30,28)(25,27,32,31)(33,35,39,46)(34,40,43,44)(36,37,42,45)(38,47,48,41);(1,3)(2,4)(5,11)(6,8)(7,10)(9,12)(13,16)(14,15)(17,20)(18,19)(21,28)(22,24)(23,26)(25,27)(29,30)(31,32)(33,36,39,42)(34,38,43,48)(35,45,46,37)(40,41,44,47);(1,4)(2,8)(3,10)(5,12)(6,7)(9,14)(11,13)(15,16)(17,19)(18,24)(20,26)(21,27)(22,23)(25,29)(28,32)(30,31)(33,37,39,45)(34,41,43,47)(35,36,46,42)(38,40,48,44);(1,5,6,15)(2,9,10,13)(3,11,8,14)(4,12,7,16)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,29)(26,32)(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=hi,c^2=gi,d^2=hi,e^2=hi,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, 8924656);