#8291636

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	#2404	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4004 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)(3,7,9,10)(5,8,13,12)(11,14,15,16)(17,18)(19,30)(20,27)(21,26)(22,32)(23,28)(24,29)(25,31)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,46)(45,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,19)(18,25)(20,28)(21,22)(23,26)(24,31)(27,32)(29,30)(33,34,37,40)(35,38,43,41)(36,39,44,46)(42,45,48,47);(3,9)(7,10)(11,15)(14,16)(17,20,24,21)(18,26,29,27)(19,28,31,22)(23,30,32,25)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48);(1,4,6,2)(3,10,9,7)(5,8,13,12)(11,14,15,16)(17,21,24,20)(18,27,29,26)(19,28,31,22)(23,30,32,25)(33,35)(34,38)(36,42)(37,43)(39,45)(40,41)(44,48)(46,47);(1,5)(2,8)(3,11)(4,12)(6,13)(7,14)(9,15)(10,16)(17,22,24,28)(18,23,29,32)(19,21,31,20)(25,26,30,27)(33,36,37,44)(34,39,40,46)(35,42,43,48)(38,45,41,47);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,23)(18,28)(19,26)(20,25)(21,30)(22,29)(24,32)(27,31)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,d^2=gi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8198832);