#8712346

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	#37832	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4874 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=h,d^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8619542);