#8463614

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

4042 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)(3,11,14,8)(4,15,13,6)(5,16,12,9)(17,18,23,22)(19,26,25,21)(20,24,28,30)(27,32,31,29)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,46)(45,48);(1,3)(2,8)(4,12)(5,13)(6,9)(7,14)(10,11)(15,16)(20,28)(24,30)(27,31)(29,32)(33,34,37,40)(35,38,43,41)(36,39,44,46)(42,45,48,47);(17,19)(18,21)(20,27)(22,26)(23,25)(24,29)(28,31)(30,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48);(1,4,7,13)(2,6,10,15)(3,12,14,5)(8,9,11,16)(17,20)(18,24)(19,27)(21,29)(22,30)(23,28)(25,31)(26,32)(33,35,37,43)(34,38,40,41)(36,42,44,48)(39,45,46,47);(1,5,7,12)(2,9,10,16)(3,13,14,4)(6,11,15,8)(17,21,23,26)(18,25,22,19)(20,29,28,32)(24,31,30,27);(1,6)(2,4)(3,9)(5,11)(7,15)(8,12)(10,13)(14,16)(17,22,23,18)(19,26,25,21)(20,30,28,24)(27,32,31,29)(33,36,37,44)(34,39,40,46)(35,42,43,48)(38,45,41,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,d^2=g,e^2=ghi,f^2=hi,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, 8370810);