#8935958

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	#2404	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4346 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,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, 8843154);