#7651067

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

4154 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)(9,10,14,13)(11,15,17,19)(12,16,20,21)(18,22,23,24)(25,26,31,35)(27,36,39,32)(28,38,37,33)(29,34,40,30);(1,2,5,6)(3,7,4,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,23)(21,24);(1,3,5,4)(2,7,6,8)(11,17)(15,19)(18,23)(22,24)(25,27,31,39)(26,32,35,36)(28,29,37,40)(30,33,34,38);(9,12)(10,16)(11,18)(13,21)(14,20)(15,22)(17,23)(19,24)(25,28)(26,33)(27,29)(30,36)(31,37)(32,34)(35,38)(39,40);(9,13,14,10)(11,19,17,15)(12,16,20,21)(18,22,23,24)(25,29)(26,34)(27,37)(28,39)(30,35)(31,40)(32,38)(33,36);(1,4,5,3)(2,8,6,7)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,23)(21,24)(25,30,31,34)(26,29,35,40)(27,38,39,33)(28,36,37,32)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gh,e^2=i,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, 7558263);