#7757965

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

4449 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7665161);