#10147824

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	#46871	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4413 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=gh,d^2=g,e^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, 10055020);