#8517604

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	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4016 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=h,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8424800);