#8325522

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	#2431	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4225 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8232718);