#8807429

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

4144 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=g,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, 8714625);