#8387987

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

4593 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8295183);