#10422790

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	#59800	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4472 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=gh,d^2=gh,e^2=h,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, 10329986);