#8648978

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	#45623	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4053 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=i,d^2=hi,e^2=gh,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, 8556174);