#7812085

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

4593 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)(9,10,15,16)(11,18,12,13)(14,17,22,23)(19,24,20,21)(25,26,31,28)(27,35,37,32)(29,38,40,33)(30,34,36,39);(1,2)(3,7)(4,8)(5,6)(9,11,15,12)(10,13,16,18)(14,20,22,19)(17,24,23,21)(25,27,31,37)(26,32,28,35)(29,36,40,30)(33,39,38,34);(10,16)(11,12)(14,22)(21,24)(26,28)(32,35)(33,38)(34,39);(1,3,5,4)(2,7,6,8)(9,12,15,11)(10,13,16,18)(14,20,22,19)(17,21,23,24)(25,28,31,26)(27,35,37,32)(29,38,40,33)(30,39,36,34);(9,13)(10,12)(11,16)(14,21)(15,18)(17,20)(19,23)(22,24)(25,29)(26,33)(27,36)(28,38)(30,37)(31,40)(32,39)(34,35);(1,4,5,3)(2,8,6,7)(9,14,15,22)(10,17,16,23)(11,19,12,20)(13,21,18,24)(25,30,31,36)(26,34,28,39)(27,29,37,40)(32,33,35,38)

Polycyclic presentation:

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

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, 7719281);