#10055631

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4696 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=gh,d^2=g,e^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=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, 9962827);