#10195506

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

4048 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)(19,27)(20,28)(21,25)(22,30)(23,24)(26,29)(31,32)(34,40)(36,46)(41,43)(44,45);(1,2,6,3)(4,9,12,7)(5,8,14,10)(11,16,13,15)(17,19,23,28)(18,20,24,27)(21,29,31,22)(25,30,32,26)(33,34)(35,41)(36,42)(37,43)(38,44)(39,40)(45,48)(46,47);(2,3)(7,9)(8,10)(15,16)(18,24)(20,27)(25,32)(26,30)(33,35,39,37)(34,41,40,43)(36,45,46,44)(38,42,48,47);(1,3,6,2)(4,7,12,9)(5,10,14,8)(11,15,13,16)(17,20,23,27)(18,19,24,28)(21,30,31,26)(22,25,29,32)(33,36)(34,42)(35,45)(37,44)(38,43)(39,46)(40,47)(41,48);(1,4)(2,7)(3,9)(5,13)(6,12)(8,16)(10,15)(11,14)(17,21,23,31)(18,25,24,32)(19,22,28,29)(20,26,27,30)(33,37,39,35)(34,43,40,41)(36,45,46,44)(38,42,48,47);(1,5)(2,8)(3,10)(4,11)(6,14)(7,15)(9,16)(12,13)(17,22)(18,26)(19,21)(20,25)(23,29)(24,30)(27,32)(28,31)(33,38,39,48)(34,44,40,45)(35,42,37,47)(36,43,46,41)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=ghi,d^2=h,e^2=g,f^2=ghi,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, 10102702);