#8745821

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	#2825	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3878 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8653017);