#8750780

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	#45094	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4187 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8657976);