#8174434

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

4634 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=hi,d^2=h,e^2=h,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8081630);