#10409726

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

4740 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=gh,d^2=gh,e^2=gh,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, 10316922);