#8479537

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	C₂×#546	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4384 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)(12,20)(15,21)(19,23)(22,24)(25,26,29,30)(27,33,34,31)(28,36,37,32)(35,38,40,39);(1,2,4,6)(3,5,8,7)(9,10)(11,18)(12,21)(13,16)(14,17)(15,20)(19,22)(23,24)(26,30)(27,34)(32,36)(35,40);(1,2)(3,5)(4,6)(7,8)(9,11,14,16)(10,13,17,18)(12,19,20,23)(15,22,21,24)(25,26)(27,31)(28,32)(29,30)(33,34)(35,38)(36,37)(39,40);(1,3)(2,5)(4,8)(6,7)(9,10)(11,13)(12,15)(14,17)(16,18)(19,22)(20,21)(23,24)(25,27,29,34)(26,31,30,33)(28,35,37,40)(32,38,36,39);(9,12)(10,15)(11,19)(13,22)(14,20)(16,23)(17,21)(18,24)(25,28)(26,32)(27,35)(29,37)(30,36)(31,38)(33,39)(34,40);(9,13)(10,16)(11,17)(12,22)(14,18)(15,23)(19,21)(20,24)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gi,d^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8386733);