#8449367

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

4153 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)(3,12)(4,9)(5,15)(6,16)(7,11)(8,14)(10,13)(18,22)(23,28)(24,29)(30,32)(33,34)(35,45)(36,41)(37,42)(38,43)(39,44)(40,46)(47,48);(1,3,7,14)(2,8,11,12)(4,13,5,6)(9,16,15,10)(19,25)(23,28)(26,31)(30,32)(35,46)(36,47)(40,45)(41,48);(1,4,7,5)(2,9,11,15)(3,13,14,6)(8,16,12,10)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,35,39,46)(34,40,44,45)(36,43,47,37)(38,41,42,48);(1,5,7,4)(2,9,11,15)(3,6,14,13)(8,16,12,10)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,36)(34,41)(35,43)(37,46)(38,45)(39,47)(40,42)(44,48);(1,6)(2,10)(3,4)(5,14)(7,13)(8,9)(11,16)(12,15)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,37,39,43)(34,42,44,38)(35,36,46,47)(40,48,45,41);(1,2)(3,8)(4,9)(5,15)(6,10)(7,11)(12,14)(13,16)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,38,39,42)(34,43,44,37)(35,41,46,48)(36,40,47,45)

Polycyclic presentation:

2^9|b^2=g,c^2=gi,d^2=g,e^2=i,f^2=h,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, 8356563);