#10442564

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	#44151	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4112 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,d^2=hi,e^2=gi,f^2=h,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, 10349760);