#10420191

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

4118 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=gh,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10327387);