#8371265

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	#2668	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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=gh,d^2=ghi,e^2=ghi,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, 8278461);