#8417327

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

4415 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)(3,7)(4,8)(5,6)(9,10,14,18)(11,12,16,15)(13,17,21,24)(19,20,23,22)(25,26,31,30)(27,34,33,29)(28,32,36,38)(35,40,39,37);(2,6)(3,8)(9,11,14,16)(10,15,18,12)(13,19,21,23)(17,22,24,20)(28,36)(32,38)(35,39)(37,40);(1,3)(2,4)(5,8)(6,7)(13,21)(17,24)(19,23)(20,22)(25,27)(26,29)(28,35)(30,34)(31,33)(32,37)(36,39)(38,40);(1,3)(2,4)(5,8)(6,7)(9,11)(10,12)(13,19)(14,16)(15,18)(17,20)(21,23)(22,24)(25,28,31,36)(26,32,30,38)(27,35,33,39)(29,37,34,40);(1,4)(2,3)(5,7)(6,8)(9,12)(10,16)(11,18)(13,20)(14,15)(17,23)(19,24)(21,22)(25,29,31,34)(26,33,30,27)(28,37,36,40)(32,39,38,35);(9,13,14,21)(10,17,18,24)(11,19,16,23)(12,20,15,22)(25,30,31,26)(27,34,33,29)(28,38,36,32)(35,40,39,37)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,d^2=hi,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8324523);