#10181309

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

4231 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)(3,10)(4,8)(5,13)(6,7)(9,15)(11,12)(14,16)(17,18,22,19)(20,23,28,25)(21,26,30,24)(27,32,29,31)(33,34,38,41)(35,37,43,39)(36,40,44,47)(42,45,48,46);(2,7)(4,10)(9,13)(12,14)(17,18,22,19)(20,25,28,23)(21,24,30,26)(27,32,29,31)(36,44)(40,47)(42,48)(45,46);(2,7)(3,8)(9,13)(11,16)(18,19)(23,25)(24,26)(31,32)(33,35,38,43)(34,39,41,37)(36,42,44,48)(40,46,47,45);(1,3)(2,4)(5,11)(6,8)(7,10)(9,12)(13,14)(15,16)(17,19,22,18)(20,23,28,25)(21,26,30,24)(27,31,29,32)(33,35)(34,37)(36,42)(38,43)(39,41)(40,45)(44,48)(46,47);(1,4)(2,8)(3,7)(5,14)(6,10)(9,11)(12,15)(13,16)(17,20)(18,23)(19,25)(21,29)(22,28)(24,32)(26,31)(27,30)(33,36,38,44)(34,40,41,47)(35,42,43,48)(37,45,39,46);(1,5,6,15)(2,9,7,13)(3,11,8,16)(4,12,10,14)(17,21)(18,24)(19,26)(20,27)(22,30)(23,31)(25,32)(28,29)(33,37)(34,35)(36,45)(38,39)(40,42)(41,43)(44,46)(47,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=ghi,d^2=i,e^2=ghi,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, 10088505);