#8824386

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

4156 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=hi,d^2=gi,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8731582);