#8271162

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	#59217	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3986 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:

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

Polycyclic presentation:

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

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, 8178358);