#10323211

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

4175 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=g,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10230407);