#7662050

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	#42882	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4285 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=g,e^2=ghi,f^2=hi,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, 7569246);