#7919962

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

4413 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)(25,26,31,28)(27,34,30,32)(29,36,38,33)(35,39,37,40);(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)(25,27,31,30)(26,32,28,34)(29,35,38,37)(33,39,36,40);(1,3,5,4)(2,7,6,8)(9,12,15,10)(11,20,19,16)(13,21,23,17)(14,22,24,18)(25,28,31,26)(27,32,30,34)(29,36,38,33)(35,39,37,40);(1,3,5,4)(2,7,6,8)(9,13)(10,17)(11,18)(12,21)(14,20)(15,23)(16,24)(19,22)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40);(1,4,5,3)(2,8,6,7)(9,14)(10,18)(11,21)(12,22)(13,16)(15,24)(17,19)(20,23)(25,30,31,27)(26,32,28,34)(29,37,38,35)(33,39,36,40)

Polycyclic presentation:

2^9|a^2=hi,c^2=ghi,d^2=gh,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, 7827158);