#10442563

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	#49806	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4352 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=hi,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, 10349759);