#7656704

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

4501 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=g,e^2=gh,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, 7563900);