#7989556

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, prime radical, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4113 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=gi,d^2=h,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7896752);