#8807475

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4262 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8714671);