#545422

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2264 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,d^2=g,e^2=fg,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dg,d^c=df

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 452618);