#10142149

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

4225 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=ghi,d^2=hi,e^2=gh,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, 10049345);