#8751669

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

4153 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)(9,10,14,13)(11,15,18,16)(12,17,20,21)(19,22,24,23)(25,26)(27,36)(28,34)(29,37)(30,32)(31,33)(35,39)(38,40);(2,3)(6,7)(9,11)(10,15)(12,19)(13,16)(14,18)(17,22)(20,24)(21,23)(25,27,31,30)(26,32,33,36)(28,37,38,35)(29,34,39,40);(1,3,5,2)(4,7,8,6)(11,18)(15,16)(19,24)(22,23)(26,33)(28,38)(32,36)(35,37);(1,4,5,8)(2,6,3,7)(9,11)(10,16)(12,19)(13,15)(14,18)(17,23)(20,24)(21,22)(25,28)(26,34)(27,35)(29,32)(30,37)(31,38)(33,40)(36,39);(9,12,14,20)(10,17,13,21)(11,19,18,24)(15,22,16,23)(25,29)(26,35)(27,34)(28,32)(30,40)(31,39)(33,37)(36,38);(9,13,14,10)(11,16,18,15)(12,17,20,21)(19,22,24,23)(25,30,31,27)(26,32,33,36)(28,35,38,37)(29,34,39,40)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gh,d^2=gh,e^2=ghi,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, 8658865);