#10324056

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

4489 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=h,e^2=gh,f^2=h,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, 10231252);