#8926949

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
prime radical	#46458	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4360 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,d^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8834145);