#8551332

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
prime radical	#55015	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3958 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)(3,7)(4,9)(5,11)(6,8)(10,12)(13,15)(14,16)(17,18,23,22)(19,24,26,21)(20,25,29,30)(27,31,32,28)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(1,3)(2,7)(4,11)(5,9)(6,12)(8,10)(13,14)(15,16)(17,19,23,26)(18,24,22,21)(20,27,29,32)(25,31,30,28)(34,38)(39,41)(40,45)(46,47);(2,8)(3,12)(9,16)(11,13)(19,26)(21,24)(27,32)(28,31)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48);(1,4)(2,9)(3,13)(5,10)(6,14)(7,15)(8,16)(11,12)(17,20)(18,25)(19,27)(21,28)(22,30)(23,29)(24,31)(26,32)(33,35,37,43)(34,39,38,41)(36,42,44,48)(40,46,45,47);(1,3)(2,10)(4,13)(5,9)(6,12)(7,8)(11,14)(15,16)(17,21)(18,19)(20,28)(22,26)(23,24)(25,27)(29,31)(30,32);(1,5)(2,11)(3,9)(4,7)(6,15)(8,13)(10,14)(12,16)(17,22,23,18)(19,21,26,24)(20,25,29,30)(27,31,32,28)(33,36,37,44)(34,40,38,45)(35,42,43,48)(39,46,41,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,d^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8458528);