#8127045

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4257 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gi,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=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, 8034241);