#10442526

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

4191 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,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, 10349722);