#8069712

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

4486 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,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, 7976908);