#7812088

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

4455 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,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, 7719284);