#7644058

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	#2815	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4158 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,d^b=dhi,e^c=ehi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7551254);