#8115878

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	#55151	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4322 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8023074);