#8073156

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

4345 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)(9,10,15,16)(11,17,13,12)(14,18,22,24)(19,23,21,20)(25,26)(27,28)(29,32)(30,33)(31,35)(34,36)(37,38)(39,40);(1,3,5,8)(2,6,4,7)(9,11,15,13)(10,12,16,17)(14,19,22,21)(18,20,24,23)(25,27,30,31)(26,28,33,35)(29,34,38,39)(32,36,37,40);(1,2,5,4)(3,6,8,7)(10,16)(11,13)(14,22)(20,23)(27,31)(28,35)(34,39)(36,40);(1,2,5,4)(3,6,8,7)(9,12,15,17)(10,11,16,13)(14,23,22,20)(18,21,24,19)(25,26)(27,28)(29,37)(30,33)(31,35)(32,38)(34,40)(36,39);(1,4,5,2)(3,7,8,6)(9,13,15,11)(10,17,16,12)(14,21,22,19)(18,23,24,20)(25,28)(26,31)(27,33)(29,36)(30,35)(32,39)(34,37)(38,40);(9,14)(10,18)(11,19)(12,20)(13,21)(15,22)(16,24)(17,23)(25,29)(26,32)(27,34)(28,36)(30,38)(31,39)(33,37)(35,40)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gh,d^2=g,e^2=g,b^a=bg,c^a=ch,e^a=ei,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, 7980352);