#8436234

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=ghi,d^2=g,e^2=g,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, 8343430);