#8665025

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	#49547	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4481 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8572221);