#8119282

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

4456 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=hi,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=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, 8026478);