#7657843

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

4436 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=hi,e^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, 7565039);