#8115390

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	#52962	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3859 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)(3,6)(4,12)(5,9)(7,11)(8,13)(10,15)(14,16)(17,18,22,19)(20,25,28,23)(21,24,29,26)(27,31,32,30)(34,38)(35,41)(36,44)(46,48);(1,3,5,11)(2,6,9,7)(4,10,13,16)(8,14,12,15)(18,19)(23,25)(24,26)(30,31)(33,34,37,38)(35,42,41,39)(36,40,44,45)(43,48,47,46);(1,2,5,9)(3,7,11,6)(4,8,13,12)(10,15,16,14)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,34)(35,39)(36,40)(37,38)(41,42)(43,46)(44,45)(47,48);(1,2)(3,7)(4,8)(5,9)(6,11)(10,15)(12,13)(14,16)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,35,37,41)(34,39,38,42)(36,43,44,47)(40,46,45,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,15)(12,14)(13,16)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,36,37,44)(34,40,38,45)(35,43,41,47)(39,46,42,48);(1,4,5,13)(2,8,9,12)(3,10,11,16)(6,14,7,15)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8022586);