#10323095

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4112 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:

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

Polycyclic presentation:

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

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, 10230291);