#10171815

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	#46787	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4325 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)(3,7)(4,5)(6,8)(9,10)(11,15)(12,16)(13,24)(14,18)(17,21)(19,20)(22,23)(25,26)(27,36)(28,33)(29,40)(30,34)(31,35)(32,39)(37,38);(1,2,5,4)(3,6,8,7)(9,11,14,20)(10,15,18,19)(12,21,23,13)(16,17,22,24)(25,27,31,39)(26,32,35,36)(28,37,40,34)(29,38,33,30);(1,3)(2,6)(4,7)(5,8)(9,10,14,18)(11,19,20,15)(12,16,23,22)(13,17,21,24)(25,28,31,40)(26,33,35,29)(27,37,39,34)(30,36,38,32);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,22)(13,17)(14,18)(16,23)(19,20)(21,24)(25,29)(26,28)(27,38)(30,39)(31,33)(32,37)(34,36)(35,40);(1,3)(2,7)(4,6)(5,8)(9,12,14,23)(10,16,18,22)(11,13,20,21)(15,24,19,17)(25,26)(27,32)(28,33)(29,40)(30,37)(31,35)(34,38)(36,39);(1,4,5,2)(3,7,8,6)(9,13)(10,17)(11,12)(14,21)(15,22)(16,19)(18,24)(20,23)(25,30,31,38)(26,34,35,37)(27,29,39,33)(28,36,40,32)

Polycyclic presentation:

2^9|b^2=gi,c^2=gh,d^2=hi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10079011);