#8757393

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

4052 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8664589);