#7954977

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

4430 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)(11,15)(12,20)(13,18)(14,16)(17,22)(19,24)(21,23)(25,26,31,30)(27,35,37,32)(28,38,39,33)(29,40,36,34);(1,3,5,8)(2,6,4,7)(9,11,14,13)(10,15,16,18)(12,19,22,21)(17,23,20,24)(25,27,31,37)(26,32,30,35)(28,36,39,29)(33,34,38,40);(1,4,5,2)(3,7,8,6)(10,16)(15,18)(17,20)(23,24)(25,28,31,39)(26,33,30,38)(27,29,37,36)(32,40,35,34);(9,10)(11,15)(12,17)(13,18)(14,16)(19,23)(20,22)(21,24)(25,28)(26,33)(27,36)(29,37)(30,38)(31,39)(32,34)(35,40);(9,12,14,22)(10,17,16,20)(11,19,13,21)(15,23,18,24)(25,29,31,36)(26,34,30,40)(27,28,37,39)(32,38,35,33);(9,13,14,11)(10,18,16,15)(12,21,22,19)(17,24,20,23)(25,30,31,26)(27,35,37,32)(28,38,39,33)(29,34,36,40)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=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, 7862173);