#8925009

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

4579 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=gh,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8832205);