#7657837

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	#39963	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4300 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)(11,16)(12,15)(13,17)(14,19)(18,22)(20,23)(21,24)(25,26)(27,29)(28,33)(30,35)(31,36)(32,34)(37,39)(38,40);(1,2,5,6)(3,7,4,8)(25,27,31,34)(26,32,36,29)(28,38,39,35)(30,37,40,33);(1,3,5,4)(2,7,6,8)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,28,31,39)(26,33,36,37)(27,35,34,38)(29,30,32,40);(9,12,13,20)(10,15,17,23)(11,18,19,24)(14,21,16,22)(25,29,31,32)(26,34,36,27)(28,30,39,40)(33,38,37,35);(1,3,5,4)(2,7,6,8)(9,11)(10,16)(12,18)(13,19)(14,17)(15,22)(20,24)(21,23)(25,30)(26,35)(27,37)(28,32)(29,39)(31,40)(33,34)(36,38);(1,4,5,3)(2,8,6,7)(9,10)(11,14)(12,15)(13,17)(16,19)(18,21)(20,23)(22,24)(25,26)(27,32)(28,33)(29,34)(30,38)(31,36)(35,40)(37,39)

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=hi,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, 7565033);