#8718569

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	#47042	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4558 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8625765);