#545527

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

2318 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)(10,14)(15,20)(16,21)(22,24)(25,26,31,34)(27,29,28,32)(30,38,39,33)(35,40,36,37);(2,4)(10,14)(11,17)(16,21)(18,23)(25,27,31,28)(26,32,34,29)(30,35,39,36)(33,40,38,37);(1,2,3,4)(5,7,8,6)(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24)(25,28,31,27)(26,29,34,32)(30,36,39,35)(33,37,38,40);(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,29)(26,28)(27,34)(30,37)(31,32)(33,36)(35,38)(39,40);(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,30,31,39)(26,33,34,38)(27,35,28,36)(29,37,32,40)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=hi,e^2=g,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, 452723);