#7645866

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

4148 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)(10,11)(13,21)(15,17)(19,24)(25,26)(27,35)(28,36)(29,33)(30,34)(31,32)(37,40)(38,39);(1,2,5,6)(3,7,4,8)(26,32)(27,36)(33,34)(37,38);(9,10,14,11)(12,15,20,17)(13,16,21,18)(19,22,24,23)(25,27)(26,28)(29,37)(30,38)(31,36)(32,35)(33,39)(34,40);(1,3,5,4)(2,7,6,8)(9,11,14,10)(12,15,20,17)(13,18,21,16)(19,22,24,23)(25,28)(26,27)(29,39)(30,40)(31,35)(32,36)(33,37)(34,38);(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,29,31,30)(26,33,32,34)(27,37,36,38)(28,39,35,40);(1,4,5,3)(2,8,6,7)(9,13)(10,16)(11,18)(12,19)(14,21)(15,22)(17,23)(20,24)(25,30,31,29)(26,34,32,33)(27,38,36,37)(28,40,35,39)

Polycyclic presentation:

2^9|b^2=gi,c^2=hi,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,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, 7553062);