#10106544

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

4253 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=hi,d^2=g,e^2=g,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, 10013740);