#8446023

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

4022 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)(17,18,21,24)(19,25,27,22)(20,23,28,30)(26,31,32,29)(34,39)(35,44)(42,46)(43,48);(3,4)(8,11)(12,14)(13,15)(17,19,21,27)(18,22,24,25)(20,26,28,32)(23,29,30,31)(33,34)(35,40)(36,42)(37,43)(38,39)(41,44)(45,46)(47,48);(1,3,7,4)(2,8,10,11)(5,12,9,14)(6,13,16,15)(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31)(33,35)(34,40)(36,43)(37,42)(38,44)(39,41)(45,48)(46,47);(1,4,7,3)(2,8,10,11)(5,14,9,12)(6,13,16,15)(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31)(33,35)(34,41)(36,43)(37,46)(38,44)(39,40)(42,47)(45,48);(1,5,7,9)(2,6,10,16)(3,12,4,14)(8,13,11,15)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,30)(31,32)(33,36,38,45)(34,42,39,46)(35,43,44,48)(37,41,47,40);(1,6)(2,9)(3,13)(4,15)(5,10)(7,16)(8,14)(11,12)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31)(33,37,38,47)(34,43,39,48)(35,42,44,46)(36,41,45,40)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=gh,d^2=gh,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, 8353219);