#8159048

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

4286 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=i,e^2=h,f^2=h,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, 8066244);