#10166463

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	#51961	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3967 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)(3,8)(4,6)(5,10)(7,9)(11,13)(12,15)(14,16)(17,18)(19,22)(20,27)(21,23)(24,28)(25,31)(26,29)(30,32)(33,34,40,38)(35,44,46,41)(36,42,37,47)(39,43,45,48);(1,3,7,4)(2,8,9,6)(5,12,14,11)(10,15,16,13)(17,19,21,26)(18,22,23,29)(20,25,28,32)(24,30,27,31)(33,35)(34,41)(36,39)(37,45)(38,44)(40,46)(42,48)(43,47);(2,9)(3,4)(10,16)(11,12)(18,23)(22,29)(24,27)(30,31)(33,36,40,37)(34,42,38,47)(35,39,46,45)(41,48,44,43);(1,4,7,3)(2,6,9,8)(5,11,14,12)(10,13,16,15)(17,18)(19,22)(20,24)(21,23)(25,30)(26,29)(27,28)(31,32)(33,37,40,36)(34,42,38,47)(35,45,46,39)(41,48,44,43);(1,5,7,14)(2,10,9,16)(3,11,4,12)(6,15,8,13)(17,19)(18,22)(20,25)(21,26)(23,29)(24,30)(27,31)(28,32)(33,38,40,34)(35,44,46,41)(36,47,37,42)(39,43,45,48);(1,6)(2,3)(4,9)(5,13)(7,8)(10,12)(11,16)(14,15)(17,20)(18,24)(19,25)(21,28)(22,30)(23,27)(26,32)(29,31)(33,39,40,45)(34,43,38,48)(35,36,46,37)(41,47,44,42)

Polycyclic presentation:

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

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, 10073659);