#8118679

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	#42997	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4733 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)(9,10,14,16)(11,18,21,15)(12,22,17,13)(19,20,23,24)(25,26)(27,34)(28,32)(29,39)(30,40)(31,33)(35,38)(36,37);(3,7)(6,8)(9,11,14,21)(10,15,16,18)(12,19,17,23)(13,20,22,24)(27,28)(32,34)(35,37)(36,38);(1,2)(3,6)(4,5)(7,8)(10,16)(12,17)(15,18)(19,23)(25,27,31,28)(26,32,33,34)(29,37,40,35)(30,36,39,38);(1,3)(2,6)(4,7)(5,8)(9,11)(10,15)(12,19)(13,20)(14,21)(16,18)(17,23)(22,24)(25,28,31,27)(26,32,33,34)(29,37,40,35)(30,38,39,36);(1,2)(3,6)(4,5)(7,8)(9,12,14,17)(10,13,16,22)(11,19,21,23)(15,20,18,24)(25,29)(26,30)(27,35)(28,37)(31,40)(32,38)(33,39)(34,36);(9,13)(10,17)(11,20)(12,16)(14,22)(15,23)(18,19)(21,24)(25,30)(26,29)(27,36)(28,38)(31,39)(32,37)(33,40)(34,35)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,d^2=i,e^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, 8025875);