#7903906

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

4624 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:

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

Polycyclic presentation:

2^9|c^2=ghi,d^2=gi,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=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, 7811102);