#10440007

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4404 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gi,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,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, 10347203);