#8850545

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

4288 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)(3,10)(4,5)(6,7)(8,11)(9,15)(12,13)(14,16)(17,18)(19,25)(20,27)(21,29)(22,23)(24,31)(26,28)(30,32)(33,34,37,41)(35,39,43,38)(36,44,45,40)(42,46,48,47);(2,7)(5,9)(8,10)(13,16)(17,19,22,27)(18,20,23,25)(21,30,31,26)(24,32,29,28)(33,34,37,41)(35,39,43,38)(36,44,45,40)(42,46,48,47);(3,11)(8,10)(12,14)(13,16)(18,23)(20,25)(24,29)(28,32)(33,35,37,43)(34,38,41,39)(36,42,45,48)(40,46,44,47);(1,3)(2,8)(4,14)(5,13)(6,11)(7,10)(9,16)(12,15)(17,20,22,25)(18,19,23,27)(21,28,31,32)(24,26,29,30)(33,35)(34,39)(36,42)(37,43)(38,41)(40,47)(44,46)(45,48);(1,4)(2,9)(3,12)(5,7)(6,15)(8,13)(10,16)(11,14)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,32)(30,31)(33,34)(35,38)(36,40)(37,41)(39,43)(42,46)(44,45)(47,48);(1,5,6,9)(2,4,7,15)(3,13,11,16)(8,14,10,12)(17,21,22,31)(18,24,23,29)(19,26,27,30)(20,28,25,32)(33,36)(34,40)(35,42)(37,45)(38,46)(39,47)(41,44)(43,48)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,d^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

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, 8757741);