#10419632

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	#46517	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3953 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,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, 10326828);