#8926643

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	#36811	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4169 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^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, 8833839);