#7736171

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	#36886	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4491 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)(9,10,14,17)(11,15,18,12)(13,16,22,24)(19,23,21,20)(25,26,30,31)(27,32,33,28)(29,34,38,37)(35,39,40,36);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,17)(13,21)(14,18)(16,20)(19,22)(23,24)(25,26)(27,32)(28,33)(29,37)(30,31)(34,38)(35,36)(39,40);(11,18)(12,15)(19,21)(20,23)(26,31)(28,32)(34,37)(36,39);(9,12,14,15)(10,11,17,18)(13,20,22,23)(16,19,24,21)(25,27,30,33)(26,32,31,28)(29,35,38,40)(34,39,37,36);(1,2,5,4)(3,6,8,7)(9,11)(10,15)(12,17)(13,19)(14,18)(16,23)(20,24)(21,22)(25,28,30,32)(26,33,31,27)(29,36,38,39)(34,40,37,35);(1,4,5,2)(3,7,8,6)(9,13)(10,16)(11,19)(12,20)(14,22)(15,23)(17,24)(18,21)(25,29,30,38)(26,34,31,37)(27,35,33,40)(28,36,32,39)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,d^2=i,e^2=g,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7643367);