#8158779

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	#45399	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3931 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)(3,9)(4,7)(5,8)(6,12)(10,14)(11,15)(13,16)(19,25)(22,28)(26,31)(29,32)(34,37)(35,43)(36,46)(42,48);(2,4)(8,9)(10,12)(15,16)(17,18,21,24)(19,22,25,28)(20,23,27,30)(26,29,31,32)(33,34,39,37)(35,44,43,40)(36,47,46,41)(38,42,45,48);(1,3,7,5)(2,8,4,9)(6,11,14,13)(10,15,12,16)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32)(33,35)(34,40)(36,45)(37,44)(38,46)(39,43)(41,48)(42,47);(1,4,7,2)(3,8,5,9)(6,12,14,10)(11,15,13,16)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32)(33,36,39,46)(34,41,37,47)(35,45,43,38)(40,48,44,42);(1,5,7,3)(2,9,4,8)(6,13,14,11)(10,16,12,15)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32)(33,37,39,34)(35,44,43,40)(36,47,46,41)(38,48,45,42);(1,6,7,14)(2,10,4,12)(3,11,5,13)(8,15,9,16)(17,20,21,27)(18,23,24,30)(19,26,25,31)(22,29,28,32)(33,38,39,45)(34,42,37,48)(35,46,43,36)(40,47,44,41)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8065975);