#8445942

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

4029 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=gh,d^2=gh,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, 8353138);