#8551030

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	#37751	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4474 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=gh,d^2=gh,e^2=gh,f^2=gi,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, 8458226);