#10103373

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	#50043	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4266 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gi,e^2=h,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, 10010569);