#9016490

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	#43421	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4166 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=i,d^2=ghi,e^2=gi,f^2=gh,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, 8923686);