#567107

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

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

2158 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=fgi,d^2=fgi,e^2=g,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dg,e^b=ef,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, 474303);