#8622068

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	#50039	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4007 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)(3,5,8,13)(4,6,9,15)(11,12,14,16)(17,18)(19,22)(20,25)(21,24)(23,28)(26,29)(27,31)(30,32)(33,34,39,42)(35,40,45,48)(36,37,38,41)(43,44,46,47);(1,3,7,8)(2,5,10,13)(4,11,9,14)(6,12,15,16)(18,19)(23,25)(24,26)(30,31)(33,35,39,45)(34,40,42,48)(36,46,38,43)(37,47,41,44);(3,8)(5,13)(11,14)(12,16)(17,18,22,19)(20,23,28,25)(21,24,29,26)(27,30,32,31)(33,36,39,38)(34,41,42,37)(35,43,45,46)(40,47,48,44);(1,4)(2,6)(3,11)(5,12)(7,9)(8,14)(10,15)(13,16)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,35)(34,40)(36,43)(37,44)(38,46)(39,45)(41,47)(42,48);(1,5,7,13)(2,8,10,3)(4,12,9,16)(6,14,15,11)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,37,39,41)(34,36,42,38)(35,44,45,47)(40,43,48,46);(1,6)(2,9)(3,12)(4,10)(5,14)(7,15)(8,16)(11,13)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,38,39,36)(34,41,42,37)(35,43,45,46)(40,44,48,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=h,d^2=g,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8529264);