#8437860

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	#2408	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4565 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=g,e^2=gi,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, 8345056);