#7652021

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

4396 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=h,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,e^d=ehi

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, 7559217);