#7657054

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4171 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)(11,22)(12,24)(13,19)(14,17)(15,20)(16,21)(18,23)(25,26)(27,31)(28,35)(29,33)(30,34)(32,38)(36,40)(37,39);(1,2,4,5)(3,6,7,8)(9,11,16,14)(10,17,21,22)(12,13,23,20)(15,18,19,24)(26,27)(32,35)(33,34)(39,40);(9,12)(10,18)(11,13)(14,20)(15,22)(16,23)(17,19)(21,24)(25,27,31,26)(28,35,38,32)(29,34,30,33)(36,40,37,39);(1,3,4,7)(2,6,5,8)(9,13,16,20)(10,19,21,15)(11,23,14,12)(17,24,22,18)(25,28,31,38)(26,32,27,35)(29,36,30,37)(33,39,34,40);(9,14,16,11)(10,17,21,22)(12,20,23,13)(15,18,19,24)(25,29,31,30)(26,33,27,34)(28,36,38,37)(32,39,35,40);(9,15,16,19)(10,20,21,13)(11,18,14,24)(12,22,23,17)(25,30,31,29)(26,34,27,33)(28,37,38,36)(32,40,35,39)

Polycyclic presentation:

2^9|b^2=gi,c^2=i,d^2=g,e^2=h,f^2=h,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, 7564250);