#8429518

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	#2537	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4031 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gi,d^2=g,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8336714);