#10247042

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

4080 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=h,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,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, 10154238);