#8746018

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×#553	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4106 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=i,d^2=i,e^2=gh,f^2=gh,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, 8653214);