#8088058

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

4484 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,d^2=i,f^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,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, 7995254);