#9064696

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	#44013	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4322 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8971892);