#8464252

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

4082 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=hi,d^2=h,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, 8371448);