#7655801

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	#42674	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4298 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)(10,15)(12,18)(16,17)(19,24)(25,26)(27,37)(28,33)(29,34)(30,35)(31,36)(32,40)(38,39);(1,2,5,6)(3,7,4,8)(9,10,14,15)(11,17,21,16)(12,13,18,23)(19,22,24,20)(25,27)(26,32)(28,38)(29,35)(30,39)(31,40)(33,34)(36,37);(9,11)(10,16)(12,19)(13,20)(14,21)(15,17)(18,24)(22,23)(25,27)(26,32)(28,38)(29,35)(30,39)(31,40)(33,34)(36,37);(1,3,5,4)(2,7,6,8)(9,12,14,18)(10,13,15,23)(11,19,21,24)(16,20,17,22)(25,28,31,35)(26,33,36,30)(27,38,40,29)(32,34,37,39);(1,3,5,4)(2,7,6,8)(9,11)(10,17)(12,19)(13,22)(14,21)(15,16)(18,24)(20,23)(25,29,31,38)(26,34,36,39)(27,35,40,28)(30,32,33,37);(1,4,5,3)(2,8,6,7)(9,13)(10,18)(11,20)(12,15)(14,23)(16,24)(17,19)(21,22)(25,30,31,33)(26,35,36,28)(27,39,40,34)(29,37,38,32)

Polycyclic presentation:

2^9|b^2=gh,d^2=g,e^2=gi,f^2=gi,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, 7562997);