#8458883

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4296 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,e^2=ghi,f^2=i,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, 8366079);