#10420156

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

4227 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)(3,4,11,7)(5,8,14,13)(10,12,16,15)(17,18)(19,25)(20,24)(21,23)(22,29)(26,30)(27,31)(28,32)(33,34,40,43)(35,44,38,41)(36,39,37,42)(45,46,47,48);(1,3,6,11)(2,7,9,4)(5,10,14,16)(8,15,13,12)(18,21)(24,25)(26,29)(31,32)(33,35,40,38)(34,41,43,44)(36,47,37,45)(39,46,42,48);(17,19,23,20)(18,24,21,25)(22,27,30,28)(26,31,29,32)(33,36,40,37)(34,42,43,39)(35,45,38,47)(41,46,44,48);(1,2)(3,7)(4,11)(5,13)(6,9)(8,14)(10,12)(15,16)(17,20,23,19)(18,25,21,24)(22,27,30,28)(26,31,29,32)(33,37,40,36)(34,39,43,42)(35,45,38,47)(41,46,44,48);(1,4,6,7)(2,3,9,11)(5,12,14,15)(8,10,13,16)(17,21,23,18)(19,25,20,24)(22,29,30,26)(27,32,28,31)(33,38,40,35)(34,41,43,44)(36,47,37,45)(39,48,42,46);(1,5)(2,8)(3,10)(4,12)(6,14)(7,15)(9,13)(11,16)(17,22)(18,26)(19,27)(20,28)(21,29)(23,30)(24,31)(25,32)(33,39)(34,36)(35,46)(37,43)(38,48)(40,42)(41,47)(44,45)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=h,d^2=h,e^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, 10327352);