#8402684

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	#37319	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4497 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)(4,6)(5,8)(9,10,15,17)(11,13,12,19)(14,16,22,24)(18,21,20,23)(26,31)(27,28)(33,38)(34,36);(1,2,4,6)(3,5,8,7)(9,10)(11,13)(12,19)(14,16)(15,17)(18,21)(20,23)(22,24)(25,26)(27,32)(28,35)(29,33)(30,31)(34,39)(36,40)(37,38);(1,2)(3,5)(4,6)(7,8)(9,11,15,12)(10,13,17,19)(14,20,22,18)(16,23,24,21)(25,27,30,28)(26,32,31,35)(29,36,37,34)(33,40,38,39);(1,3)(2,5)(4,8)(6,7)(9,12,15,11)(10,13,17,19)(14,20,22,18)(16,21,24,23)(25,26)(27,32)(28,35)(29,33)(30,31)(34,39)(36,40)(37,38);(9,13)(10,12)(11,17)(14,21)(15,19)(16,20)(18,24)(22,23)(25,28,30,27)(26,32,31,35)(29,36,37,34)(33,39,38,40);(9,14,15,22)(10,16,17,24)(11,18,12,20)(13,21,19,23)(25,29,30,37)(26,33,31,38)(27,34,28,36)(32,39,35,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=gi,d^2=ghi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8309880);