#10451676

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	#43101	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4460 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)(9,10,15,12)(11,19,21,16)(13,17,20,22)(14,18,24,23)(26,30)(28,36)(31,33)(34,37);(9,11,15,21)(10,16,12,19)(13,14,20,24)(17,23,22,18)(26,30)(31,33)(32,38)(39,40);(1,2)(3,6)(4,5)(7,8)(25,26)(27,31)(28,32)(29,30)(33,35)(34,39)(36,38)(37,40);(3,7)(6,8)(9,12,15,10)(11,16,21,19)(13,22,20,17)(14,23,24,18)(25,26)(27,33)(28,32)(29,30)(31,35)(34,40)(36,38)(37,39);(1,3)(2,6)(4,7)(5,8)(9,13)(10,17)(11,14)(12,22)(15,20)(16,23)(18,19)(21,24)(25,27)(26,31)(28,37)(29,35)(30,33)(32,40)(34,36)(38,39);(1,2)(3,6)(4,5)(7,8)(9,14)(10,18)(11,20)(12,23)(13,21)(15,24)(16,17)(19,22)(25,28)(26,32)(27,34)(29,36)(30,38)(31,39)(33,40)(35,37)

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10358872);