#10158644

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

4052 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)(3,10)(4,8)(5,13)(6,7)(9,15)(11,12)(14,16)(17,18,23,25)(19,24,27,31)(20,22,29,32)(21,26,28,30)(33,34)(35,38)(36,43)(37,40)(39,44)(41,47)(42,46)(45,48);(2,7)(4,10)(9,13)(12,14)(17,19,23,27)(18,24,25,31)(20,28,29,21)(22,30,32,26)(33,35,37,42)(34,38,40,46)(36,41,44,48)(39,45,43,47);(2,7)(3,8)(9,13)(11,16)(18,25)(19,27)(20,29)(26,30)(33,34,37,40)(35,38,42,46)(36,39,44,43)(41,45,48,47);(1,3)(2,4)(5,11)(6,8)(7,10)(9,12)(13,14)(15,16)(17,20)(18,22)(19,28)(21,27)(23,29)(24,30)(25,32)(26,31)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,4)(2,8)(3,7)(5,14)(6,10)(9,11)(12,15)(13,16)(17,21,23,28)(18,26,25,30)(19,29,27,20)(22,24,32,31)(33,35)(34,38)(36,41)(37,42)(39,45)(40,46)(43,47)(44,48);(1,5,6,15)(2,9,7,13)(3,11,8,16)(4,12,10,14)(17,22)(18,20)(19,30)(21,31)(23,32)(24,28)(25,29)(26,27)(33,36)(34,39)(35,41)(37,44)(38,45)(40,43)(42,48)(46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=ghi,e^2=i,f^2=g,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, 10065840);