#8750314

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	#39464	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4668 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=gi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8657510);