#10201949

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	#54863	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4378 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gh,d^2=gi,e^2=gh,f^2=ghi,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, 10109145);