#7656554

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	#41012	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4285 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)(11,16,21,15)(12,13,17,23)(19,22,24,20)(25,26)(27,30)(28,34)(29,37)(31,33)(32,36)(35,38)(39,40);(1,2,5,6)(3,7,4,8)(25,27,32,31)(26,33,36,30)(28,37,39,35)(29,34,38,40);(1,3,5,4)(2,7,6,8)(9,11)(10,15)(12,19)(13,20)(14,21)(16,18)(17,24)(22,23)(25,28,32,39)(26,34,36,40)(27,35,31,37)(29,30,38,33);(1,3,5,4)(2,7,6,8)(9,12,14,17)(10,13,18,23)(11,19,21,24)(15,20,16,22)(25,29,32,38)(26,35,36,37)(27,34,31,40)(28,30,39,33);(1,4,5,3)(2,8,6,7)(9,11)(10,16)(12,19)(13,22)(14,21)(15,18)(17,24)(20,23)(25,30)(26,27)(28,38)(29,39)(31,36)(32,33)(34,35)(37,40);(9,13)(10,17)(11,20)(12,18)(14,23)(15,24)(16,19)(21,22)(25,31,32,27)(26,33,36,30)(28,37,39,35)(29,40,38,34)

Polycyclic presentation:

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

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, 7563750);