#8241364

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	#45373	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4365 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:

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

Polycyclic presentation:

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

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, 8148560);