#10420238

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

4505 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=g,d^2=g,e^2=h,f^2=g,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, 10327434);