#8926081

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

4572 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)(4,5)(6,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(27,28)(29,37)(31,34)(32,39);(2,5)(6,7)(10,14)(15,17)(16,21)(22,23)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,38)(37,39);(1,2)(3,6)(4,5)(7,8)(12,20)(16,21)(18,24)(22,23)(25,27,30,28)(26,31,33,34)(29,36,37,35)(32,40,39,38);(1,3,4,8)(2,6,5,7)(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,28,30,27)(26,31,33,34)(29,36,37,35)(32,38,39,40);(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,26)(27,31)(28,34)(29,32)(30,33)(35,38)(36,40)(37,39);(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,29,30,37)(26,32,33,39)(27,35,28,36)(31,38,34,40)

Polycyclic presentation:

2^9|c^2=h,d^2=gh,f^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, 8833277);