#8445754

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

4013 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=h,e^2=gi,f^2=g,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, 8352950);