#10451582

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

3946 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gh,d^2=gi,e^2=hi,f^2=i,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, 10358778);