#7645978

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

4588 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=g,e^2=i,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, 7553174);