#8688224

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4072 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)(3,7)(4,5)(6,8)(9,10)(11,21)(12,17)(13,18)(14,19)(15,20)(16,22)(23,24)(25,26,30,34)(27,32,28,31)(29,37,38,33)(35,39,36,40);(2,5)(6,7)(9,11,15,22)(10,16,20,21)(12,13,19,23)(14,18,17,24)(25,26)(27,32)(28,31)(29,33)(30,34)(35,40)(36,39)(37,38);(1,2)(3,6)(4,5)(7,8)(9,12,15,19)(10,17,20,14)(11,13,22,23)(16,24,21,18)(25,27,30,28)(26,31,34,32)(29,35,38,36)(33,39,37,40);(1,3)(2,6)(4,8)(5,7)(9,13)(10,18)(11,19)(12,22)(14,21)(15,23)(16,17)(20,24)(25,28,30,27)(26,32,34,31)(29,36,38,35)(33,40,37,39);(9,13)(10,18)(11,19)(12,22)(14,21)(15,23)(16,17)(20,24)(25,29,30,38)(26,33,34,37)(27,35,28,36)(31,39,32,40);(9,14,15,17)(10,19,20,12)(11,18,22,24)(13,16,23,21)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=ghi,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,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, 8595420);