#7658605

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

4305 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)(9,10,15,16)(11,17,21,23)(12,14,20,18)(13,19,22,24)(25,26)(27,30)(28,36)(29,34)(31,35)(32,37)(33,39)(38,40);(1,2)(3,6)(4,7)(5,8)(10,16)(11,21)(12,20)(19,24)(25,27)(26,32)(28,34)(29,33)(30,35)(31,37)(36,38)(39,40);(1,3,5,4)(2,6,8,7)(9,11,15,21)(10,17,16,23)(12,13,20,22)(14,19,18,24)(25,28)(26,33)(27,34)(29,32)(30,38)(31,39)(35,36)(37,40);(1,3,5,4)(2,6,8,7)(9,12,15,20)(10,18,16,14)(11,13,21,22)(17,24,23,19)(25,29,31,38)(26,34,35,40)(27,36,37,33)(28,32,39,30);(1,4,5,3)(2,7,8,6)(9,13)(10,19)(11,20)(12,21)(14,23)(15,22)(16,24)(17,18)(25,30,31,32)(26,27,35,37)(28,38,39,29)(33,34,36,40);(9,14,15,18)(10,20,16,12)(11,19,21,24)(13,17,22,23)(25,27)(26,30)(28,34)(29,36)(31,37)(32,35)(33,38)(39,40)

Polycyclic presentation:

2^9|a^2=gh,c^2=gh,d^2=g,e^2=gi,f^2=i,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, 7565801);