#8528968

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4398 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8436164);