#8323188

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	#42539	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4428 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,e^2=g,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, 8230384);