#8662910

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	#55553	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4433 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8570106);