#8087986

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

4408 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)(10,15)(12,22)(16,18)(19,23)(25,26,31,35)(27,36,28,32)(29,30,34,33)(37,40,39,38);(1,2)(3,5)(4,6)(7,8)(9,10,14,15)(11,18,21,16)(12,17,22,13)(19,20,23,24)(27,28)(32,36)(37,39)(38,40);(10,15)(12,22)(16,18)(19,23)(25,27,31,28)(26,32,35,36)(29,39,34,37)(30,40,33,38);(9,11)(10,16)(12,23)(13,24)(14,21)(15,18)(17,20)(19,22)(25,28,31,27)(26,32,35,36)(29,39,34,37)(30,38,33,40);(9,12)(10,17)(11,19)(13,15)(14,22)(16,24)(18,20)(21,23)(25,29,31,34)(26,33,35,30)(27,37,28,39)(32,40,36,38);(1,3,4,7)(2,5,6,8)(9,13,14,17)(10,12,15,22)(11,20,21,24)(16,19,18,23)(25,30,31,33)(26,34,35,29)(27,38,28,40)(32,39,36,37)

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=h,d^2=h,e^2=h,f^2=gh,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, 7995182);