#8073132

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

4342 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)(11,20)(12,16)(13,18)(14,19)(15,17)(21,22)(23,24)(25,26)(27,28)(29,32)(30,33)(31,35)(34,36)(37,38)(39,40);(1,3,5,8)(2,6,4,7)(9,11,15,12)(10,16,17,20)(13,21,19,23)(14,22,18,24)(25,27,30,31)(26,28,33,35)(29,34,38,39)(32,36,37,40);(1,2,5,4)(3,6,8,7)(10,17)(14,18)(16,20)(22,24)(27,31)(28,35)(34,39)(36,40);(1,2,5,4)(3,6,8,7)(9,12,15,11)(10,16,17,20)(13,21,19,23)(14,24,18,22)(25,26)(27,28)(29,37)(30,33)(31,35)(32,38)(34,40)(36,39);(1,2,5,4)(3,6,8,7)(9,13,15,19)(10,18,17,14)(11,21,12,23)(16,24,20,22)(25,28)(26,31)(27,33)(29,36)(30,35)(32,39)(34,37)(38,40);(1,4,5,2)(3,7,8,6)(9,14,15,18)(10,19,17,13)(11,22,12,24)(16,23,20,21)(25,29)(26,32)(27,34)(28,36)(30,38)(31,39)(33,37)(35,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gh,d^2=g,e^2=g,f^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, 7980328);