#10155812

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

4423 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=ghi,d^2=g,e^2=gi,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, 10063008);