#8438503

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	#35226	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4586 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=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, 8345699);