#8359858

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3989 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,d^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=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, 8267054);