#8551137

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

4108 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,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, 8458333);