#10440019

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

4371 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gi,e^2=gi,f^2=h,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, 10347215);