#8755971

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	#47170	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4340 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=h,e^2=gi,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, 8663167);