#7645461

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

4539 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=g,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, 7552657);