#10336063

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	#46413	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4358 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=ghi,d^2=g,e^2=g,f^2=g,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, 10243259);