#7657889

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

4401 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=hi,e^2=gh,f^2=h,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, 7565085);