#7658234

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

4608 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)(12,20)(15,21)(18,24)(22,23)(25,26)(27,29)(28,32)(30,33)(31,35)(34,37)(36,39)(38,40);(1,3,5,8)(2,6,4,7)(9,10)(11,16)(12,21)(13,17)(14,19)(15,20)(18,22)(23,24)(28,36)(32,39)(34,40)(37,38);(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,24)(21,23)(25,27)(26,31)(28,34)(29,33)(30,35)(32,38)(36,40)(37,39);(9,12)(10,15)(11,18)(13,20)(14,22)(16,23)(17,21)(19,24)(25,28)(26,32)(27,34)(29,37)(30,36)(31,38)(33,39)(35,40);(9,11)(10,16)(12,18)(13,19)(14,17)(15,23)(20,24)(21,22)(25,29,30,31)(26,27,33,35)(28,37,36,38)(32,34,39,40);(1,4,5,2)(3,7,8,6)(9,10)(11,14)(12,15)(13,17)(16,19)(18,22)(20,21)(23,24)(25,27)(26,29)(28,34)(30,35)(31,33)(32,37)(36,40)(38,39)

Polycyclic presentation:

2^9|a^2=g,b^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7565430);