#7656841

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4167 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)(9,10,16,20)(11,21,22,17)(12,23,14,18)(13,15,19,24)(25,26,29,30)(27,33,35,31)(28,32,36,37)(34,39,40,38);(1,2)(3,7)(4,8)(5,6)(9,11)(10,17)(12,13)(14,19)(15,18)(16,22)(20,21)(23,24)(26,30)(31,33)(32,37)(38,39);(9,12,16,14)(10,18,20,23)(11,13,22,19)(15,21,24,17)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40);(1,3,5,4)(2,7,6,8)(9,13,16,19)(10,15,20,24)(11,12,22,14)(17,18,21,23)(25,27,29,35)(26,31,30,33)(28,34,36,40)(32,38,37,39);(1,4,5,3)(2,8,6,7)(9,14,16,12)(10,18,20,23)(11,19,22,13)(15,21,24,17);(9,15)(10,19)(11,18)(12,21)(13,20)(14,17)(16,24)(22,23)(25,28,29,36)(26,32,30,37)(27,34,35,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=h,c^2=hi,d^2=g,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,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, 7564037);