#8635454

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	#38132	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4113 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=h,d^2=g,e^2=gh,f^2=hi,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, 8542650);