#8446000

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

4007 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=gh,d^2=h,e^2=i,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, 8353196);