#8348952

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	#2847	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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=g,d^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=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, 8256148);