#8088337

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

4471 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)(10,15)(12,18)(16,19)(20,23)(25,26,31,32)(27,29,28,33)(30,34,38,39)(35,37,36,40);(1,2)(3,7)(4,8)(5,6)(9,10)(11,19)(12,17)(13,18)(14,15)(16,22)(20,21)(23,24)(26,32)(27,28)(34,39)(35,36);(10,15)(12,18)(16,19)(20,23)(25,27,31,28)(26,33,32,29)(30,36,38,35)(34,37,39,40);(1,3,5,4)(2,7,6,8)(9,11)(10,16)(12,23)(13,24)(14,22)(15,19)(17,21)(18,20)(25,28,31,27)(26,33,32,29)(30,35,38,36)(34,37,39,40);(1,3,5,4)(2,7,6,8)(9,12,14,18)(10,17,15,13)(11,20,22,23)(16,24,19,21)(25,29,31,33)(26,27,32,28)(30,37,38,40)(34,35,39,36);(1,4,5,3)(2,8,6,7)(9,13,14,17)(10,18,15,12)(11,21,22,24)(16,23,19,20)(25,30,31,38)(26,34,32,39)(27,35,28,36)(29,37,33,40)

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=ghi,e^2=g,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, 7995533);