#10412339

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3972 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)(3,9)(4,8)(5,7)(6,10)(11,16)(12,15)(13,14)(17,18,23,26)(19,24,30,32)(20,25,28,29)(21,31,27,22)(33,34,40,43)(35,38,42,48)(36,45,47,41)(37,39,46,44);(1,3,7,4)(2,8,5,9)(6,11,14,12)(10,15,13,16)(17,19)(18,24)(20,21)(22,29)(23,30)(25,31)(26,32)(27,28)(33,35,40,42)(34,38,43,48)(36,44,47,39)(37,41,46,45);(1,3,7,4)(2,9,5,8)(6,11,14,12)(10,16,13,15)(17,20)(18,25)(19,27)(21,30)(22,24)(23,28)(26,29)(31,32)(33,36,40,47)(34,41,43,45)(35,44,42,39)(37,38,46,48);(1,4,7,3)(2,9,5,8)(6,11,14,12)(10,15,13,16)(17,20)(18,25)(19,21)(22,32)(23,28)(24,31)(26,29)(27,30)(33,37)(34,39)(35,41)(36,38)(40,46)(42,45)(43,44)(47,48);(1,5,7,2)(3,8,4,9)(6,13,14,10)(11,15,12,16)(17,21)(18,22)(19,28)(20,30)(23,27)(24,25)(26,31)(29,32)(33,38)(34,42)(35,43)(36,46)(37,47)(39,41)(40,48)(44,45);(1,6,7,14)(2,10,5,13)(3,11,4,12)(8,15,9,16)(17,22)(18,21)(19,29)(20,24)(23,31)(25,30)(26,27)(28,32)(33,39)(34,37)(35,36)(38,41)(40,44)(42,47)(43,46)(45,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gh,d^2=g,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, 10319535);