#8900058

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

4666 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gh,d^2=gh,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=egi

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, 8807254);