#7663138

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	#42103	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4416 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)(11,15,18,22)(12,13,16,21)(19,20,23,24)(25,26,30,32)(27,34,37,31)(28,38,39,33)(29,35,40,36);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,19)(13,20)(14,18)(16,23)(17,22)(21,24)(25,27)(26,31)(28,36)(29,33)(30,37)(32,34)(35,39)(38,40);(1,2,5,4)(3,6,8,7)(11,18)(15,22)(19,23)(20,24)(26,32)(28,39)(31,34)(35,36);(1,2,5,4)(3,6,8,7)(9,11)(10,15)(12,19)(13,20)(14,18)(16,23)(17,22)(21,24)(25,28)(26,33)(27,36)(29,31)(30,39)(32,38)(34,40)(35,37);(1,4,5,2)(3,7,8,6)(9,12)(10,13)(11,19)(14,16)(15,20)(17,21)(18,23)(22,24)(25,27)(26,34)(28,36)(29,38)(30,37)(31,32)(33,40)(35,39);(9,13)(10,16)(11,20)(12,17)(14,21)(15,23)(18,24)(19,22)(25,29,30,40)(26,35,32,36)(27,33,37,38)(28,31,39,34)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gh,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^c=dhi,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, 7570334);