#8069345

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

4446 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=gh,d^2=gh,e^2=h,f^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7976541);