#8648965

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3945 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=i,d^2=ghi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8556161);