#7645857

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	#42232	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4226 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:

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

Polycyclic presentation:

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

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, 7553053);