#7656855

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

4199 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)(9,10)(11,18)(12,23)(13,17)(14,19)(15,22)(16,21)(20,24)(25,26,30,34)(27,35,36,31)(28,32,29,33)(37,40,38,39);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,20)(13,21)(14,22)(16,24)(17,23)(18,19)(27,36)(31,35)(37,38)(39,40);(9,12,14,21)(10,16,19,23)(11,20,22,13)(15,24,18,17)(25,27)(26,31)(28,37)(29,38)(30,36)(32,39)(33,40)(34,35);(1,3,4,7)(2,5,6,8)(9,13,14,20)(10,17,19,24)(11,21,22,12)(15,23,18,16)(25,26)(27,31)(28,32)(29,33)(30,34)(35,36)(37,39)(38,40);(9,11)(10,18)(12,20)(13,21)(14,22)(15,19)(16,17)(23,24)(25,28,30,29)(26,32,34,33)(27,37,36,38)(31,39,35,40);(9,10)(11,15)(12,16)(13,17)(14,19)(18,22)(20,24)(21,23)(25,29,30,28)(26,33,34,32)(27,38,36,37)(31,40,35,39)

Polycyclic presentation:

2^9|a^2=gh,c^2=hi,d^2=g,e^2=i,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, 7564051);