#8437764

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	#58003	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4418 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)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(26,29)(27,28)(34,37)(35,36);(1,2,4,6)(3,5,8,7)(9,11,14,17)(10,15,18,13)(12,19,21,23)(16,22,24,20)(25,26,31,29)(27,33,28,32)(30,34,38,37)(35,40,36,39);(1,2)(3,5)(4,6)(7,8)(9,11)(10,13)(12,19)(14,17)(15,18)(16,20)(21,23)(22,24)(25,27,31,28)(26,32,29,33)(30,35,38,36)(34,39,37,40);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,28,31,27)(26,33,29,32)(30,36,38,35)(34,40,37,39);(9,12,14,21)(10,16,18,24)(11,19,17,23)(13,20,15,22)(25,29,31,26)(27,33,28,32)(30,37,38,34)(35,40,36,39);(9,13,14,15)(10,17,18,11)(12,20,21,22)(16,23,24,19)(25,30,31,38)(26,34,29,37)(27,35,28,36)(32,39,33,40)

Polycyclic presentation:

2^9|b^2=i,c^2=ghi,d^2=g,e^2=h,f^2=h,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, 8344960);