#8926155

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

4485 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)(11,15)(12,16)(13,17)(14,19)(18,22)(20,23)(21,24)(25,26,31,34)(27,29,32,28)(30,38,39,33)(35,36,40,37);(1,2,4,6)(3,5,8,7)(9,11,13,19)(10,14,17,15)(12,18,20,24)(16,21,23,22)(25,27)(26,29)(28,34)(30,35)(31,32)(33,37)(36,38)(39,40);(1,2)(3,5)(4,6)(7,8)(12,20)(16,23)(18,24)(21,22)(25,28,31,29)(26,32,34,27)(30,37,39,36)(33,35,38,40);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,16)(13,17)(15,19)(18,21)(20,23)(22,24)(25,27)(26,28)(29,34)(30,35)(31,32)(33,36)(37,38)(39,40);(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,29,31,28)(26,32,34,27)(30,37,39,36)(33,40,38,35);(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,30,31,39)(26,33,34,38)(27,35,32,40)(28,36,29,37)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,e^2=h,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, 8833351);