#7812094

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

4390 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,e^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, 7719290);