#8612914

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

4481 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)(11,18)(12,20)(13,15)(14,16)(17,22)(19,23)(21,24)(25,26,30,32)(27,33,35,28)(29,31,38,37)(34,39,40,36);(1,2)(3,5)(4,6)(7,8)(9,11,15,14)(10,16,13,18)(12,21,22,19)(17,24,20,23)(25,27,30,35)(26,28,32,33)(29,34,38,40)(31,36,37,39);(2,3)(6,7)(10,13)(16,18)(17,20)(23,24)(25,28,30,33)(26,27,32,35)(29,36,38,39)(31,34,37,40);(4,8)(6,7)(9,12)(10,17)(11,19)(13,20)(14,21)(15,22)(16,23)(18,24)(25,26)(27,28)(29,37)(30,32)(31,38)(33,35)(34,39)(36,40);(1,3,5,2)(4,7,8,6)(9,13,15,10)(11,16,14,18)(12,20,22,17)(19,23,21,24)(25,28)(26,27)(29,36)(30,33)(31,34)(32,35)(37,40)(38,39);(1,4)(2,6)(3,7)(5,8)(9,14,15,11)(10,16,13,18)(12,19,22,21)(17,24,20,23)(25,29,30,38)(26,31,32,37)(27,34,35,40)(28,36,33,39)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8520110);