#10219839

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
prime radical	#38521	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4143 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10127035);