#7714904

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

4313 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7622100);