#7652587

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	#39126	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4305 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)(9,10,14,15)(11,17,19,13)(12,16,22,23)(18,24,20,21)(25,26,30,28)(27,31,33,34)(29,32,37,36)(35,38,39,40);(1,2,5,6)(3,7,4,8)(9,11)(10,13)(12,18)(14,19)(15,17)(16,21)(20,22)(23,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(1,3,5,4)(2,7,6,8)(10,15)(13,17)(16,23)(21,24)(27,33)(31,34)(35,39)(38,40);(9,11)(10,13)(12,20)(14,19)(15,17)(16,24)(18,22)(21,23)(25,28,30,26)(27,34,33,31)(29,32,37,36)(35,38,39,40);(1,3,5,4)(2,7,6,8)(9,12,14,22)(10,16,15,23)(11,18,19,20)(13,21,17,24)(25,29,30,37)(26,32,28,36)(27,35,33,39)(31,38,34,40);(1,4,5,3)(2,8,6,7)(9,13)(10,11)(12,21)(14,17)(15,19)(16,18)(20,23)(22,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gh,d^2=i,e^2=gi,f^2=gh,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, 7559783);