#8745850

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

4096 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=i,d^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8653046);