#8696663

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	#60032	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4147 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=hi,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8603859);