#8925205

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

4794 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)(3,7)(4,6)(5,8)(9,10,15,17)(11,18,12,16)(13,14,23,24)(19,22,21,20)(25,26)(27,31)(28,32)(29,33)(30,35)(34,38)(36,39)(37,40);(1,2,4,6)(3,5,8,7)(9,11,15,12)(10,16,17,18)(13,19,23,21)(14,20,24,22)(25,27,29,35)(26,30,33,31)(28,34,36,40)(32,37,39,38);(1,2)(3,5)(4,6)(7,8)(10,17)(14,24)(16,18)(20,22)(28,36)(32,39)(34,40)(37,38);(1,3)(2,5)(4,8)(6,7)(9,12,15,11)(10,16,17,18)(13,19,23,21)(14,22,24,20)(25,26)(27,30)(28,32)(29,33)(31,35)(34,37)(36,39)(38,40);(9,13)(10,14)(11,19)(12,21)(15,23)(16,20)(17,24)(18,22)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(9,14)(10,13)(11,20)(12,22)(15,24)(16,19)(17,23)(18,21)(25,28)(26,32)(27,34)(29,36)(30,37)(31,38)(33,39)(35,40)

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=h,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, 8832401);