#8745907

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	#52328	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4007 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:

(1,2,5,6)(3,9,11,7)(4,8,14,12)(10,16,13,15)(17,18,23,22)(19,26,21,24)(20,25,28,30)(27,32,29,31)(33,34,39,44)(35,42,37,40)(36,41,47,46)(38,43,48,45);(1,2,5,6)(3,7,11,9)(4,8,14,12)(10,15,13,16)(17,19,23,21)(18,24,22,26)(20,27,28,29)(25,31,30,32)(33,35,39,37)(34,40,44,42)(36,43,47,45)(38,46,48,41);(2,6)(7,9)(8,12)(15,16)(20,28)(25,30)(27,29)(31,32)(33,35,39,37)(34,40,44,42)(36,43,47,45)(38,46,48,41);(1,2)(3,7)(4,12)(5,6)(8,14)(9,11)(10,16)(13,15)(17,20)(18,25)(19,27)(21,29)(22,30)(23,28)(24,31)(26,32)(33,36,39,47)(34,41,44,46)(35,45,37,43)(38,40,48,42);(1,3)(2,7)(4,13)(5,11)(6,9)(8,16)(10,14)(12,15)(17,21,23,19)(18,24,22,26)(20,29,28,27)(25,31,30,32)(33,37,39,35)(34,42,44,40)(36,43,47,45)(38,46,48,41);(1,4,5,14)(2,8,6,12)(3,10,11,13)(7,15,9,16)(17,22,23,18)(19,26,21,24)(20,25,28,30)(27,31,29,32)(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=hi,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, 8653103);