#8075487

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

4422 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=ghi,d^2=gh,e^2=h,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, 7982683);