#8750875

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×D₄²	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4379 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=gh,d^2=gi,e^2=g,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, 8658071);