#10412330

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	#46697	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4102 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,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, 10319526);