#8464312

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, prime radical, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3875 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,d^2=g,e^2=ghi,f^2=gi,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, 8371508);