#7658323

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
prime radical	#42476	C₂
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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,e^2=gi,f^2=gi,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, 7565519);