#10320799

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

4510 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,d^2=g,f^2=g,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, 10227995);