#8656193

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	#45647	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4186 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8563389);