#8533570

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	#54839	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4176 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:

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

Polycyclic presentation:

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

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, 8440766);