#7747293

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

4662 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)(11,17)(15,21)(18,23)(22,24)(25,26,30,33)(27,28,35,31)(29,32,38,40)(34,36,39,37);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,27,30,35)(26,31,33,28)(29,34,38,39)(32,37,40,36);(10,14)(15,21)(16,19)(22,24)(25,28,30,31)(26,27,33,35)(29,37,38,36)(32,39,40,34);(1,3,5,4)(2,7,6,8)(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23)(25,28)(26,27)(29,36)(30,31)(32,34)(33,35)(37,38)(39,40);(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,26)(27,28)(29,32)(30,33)(31,35)(34,36)(37,39)(38,40);(1,4,5,3)(2,8,6,7)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,29,30,38)(26,32,33,40)(27,34,35,39)(28,36,31,37)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,e^b=ei,d^c=dh,f^c=fi

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, 7654489);