#7658842

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	#42952	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4352 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)(9,10,15,14)(11,18,13,16)(12,17,20,22)(19,24,21,23)(25,26)(27,36)(28,34)(29,32)(30,35)(31,33)(37,40)(38,39);(1,2,5,6)(3,7,4,8)(12,20)(17,22)(19,21)(23,24)(25,27,32,31)(26,33,29,36)(28,38,30,37)(34,40,35,39);(9,11,15,13)(10,16,14,18)(12,19,20,21)(17,23,22,24)(25,28,32,30)(26,34,29,35)(27,37,31,38)(33,39,36,40);(1,3,5,4)(2,7,6,8)(9,12)(10,17)(11,19)(13,21)(14,22)(15,20)(16,23)(18,24)(25,29,32,26)(27,33,31,36)(28,35,30,34)(37,39,38,40);(1,3,5,4)(2,7,6,8)(9,13,15,11)(10,16,14,18)(12,21,20,19)(17,23,22,24)(25,30,32,28)(26,35,29,34)(27,37,31,38)(33,39,36,40);(1,4,5,3)(2,8,6,7)(9,14,15,10)(11,18,13,16)(12,22,20,17)(19,24,21,23)(25,31,32,27)(26,33,29,36)(28,38,30,37)(34,39,35,40)

Polycyclic presentation:

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

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, 7566038);