#8799171

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

4134 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=gi,e^2=gi,f^2=gh,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, 8706367);