#7663095

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	#42685	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4252 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gh,d^2=gh,e^2=i,f^2=i,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, 7570291);