#7643232

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	#39962	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4364 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)(13,21)(18,22)(19,24)(20,23)(27,28)(30,32)(35,36)(37,39);(1,2,4,5)(3,6,7,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,24)(21,22)(25,26,31,34)(27,32,28,30)(29,33,38,40)(35,39,36,37);(10,15)(12,16)(18,22)(20,23)(25,27,31,28)(26,32,34,30)(29,36,38,35)(33,37,40,39);(9,11,14,17)(10,16,15,12)(13,19,21,24)(18,23,22,20)(25,28,31,27)(26,30,34,32)(29,36,38,35)(33,37,40,39);(9,12,14,16)(10,17,15,11)(13,20,21,23)(18,24,22,19)(25,29,31,38)(26,33,34,40)(27,35,28,36)(30,37,32,39);(1,3,4,7)(2,6,5,8)(9,13)(10,18)(11,19)(12,20)(14,21)(15,22)(16,23)(17,24)(25,30)(26,27)(28,34)(29,37)(31,32)(33,35)(36,40)(38,39)

Polycyclic presentation:

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

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, 7550428);