#7888946

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

4470 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,d^2=gh,e^2=gi,f^2=hi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7796142);