#8464040

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4149 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=ghi,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,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, 8371236);