#10457905

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

4210 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10365101);