#10195887

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

4037 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=hi,d^2=gh,e^2=gh,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, 10103083);