#10181303

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

4084 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=ghi,d^2=i,e^2=hi,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, 10088499);