#10166264

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	#46777	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4069 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)(3,8)(4,5)(6,14)(7,9)(10,16)(11,13)(12,15)(17,18,21,23)(19,24,27,22)(20,28,30,25)(26,31,29,32)(34,38)(36,44)(39,41)(42,48);(1,3,7,4)(2,8,9,5)(6,11,16,12)(10,15,14,13)(17,19,21,27)(18,22,23,24)(20,26,30,29)(25,31,28,32)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(2,9)(3,4)(10,14)(11,12)(18,23)(22,24)(25,28)(31,32)(33,34,37,38)(35,39,43,41)(36,40,44,45)(42,46,48,47);(1,4,7,3)(2,5,9,8)(6,12,16,11)(10,13,14,15)(17,18)(19,22)(20,25)(21,23)(24,27)(26,31)(28,30)(29,32)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48);(1,5)(2,4)(3,9)(6,15)(7,8)(10,11)(12,14)(13,16)(17,19)(18,24)(20,29)(21,27)(22,23)(25,31)(26,30)(28,32)(33,35,37,43)(34,39,38,41)(36,42,44,48)(40,46,45,47);(1,6,7,16)(2,10,9,14)(3,11,4,12)(5,13,8,15)(17,20)(18,25)(19,26)(21,30)(22,31)(23,28)(24,32)(27,29)(33,36)(34,40)(35,42)(37,44)(38,45)(39,46)(41,47)(43,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=ghi,d^2=gh,e^2=ghi,f^2=gh,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, 10073460);