#8656218

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	#2541	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4081 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:

(1,2,7,4)(3,8,5,9)(6,10,14,12)(11,15,13,16)(17,18,22,25)(19,26,21,23)(20,24,29,32)(27,31,28,30)(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)(17,19,22,21)(18,23,25,26)(20,28,29,27)(24,31,32,30)(33,35,40,39)(34,41,45,46)(36,47,37,44)(38,43,48,42);(3,5)(8,9)(11,13)(15,16)(20,29)(24,32)(27,28)(30,31)(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)(17,18)(19,23)(20,24)(21,26)(22,25)(27,30)(28,31)(29,32)(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)(18,24)(19,27)(21,28)(22,29)(23,30)(25,32)(26,31)(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)(17,21,22,19)(18,23,25,26)(20,28,29,27)(24,30,32,31)(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=h,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, 8563414);