#557284

order	512
nilpotency class	2
isoclinism	isoclinic class of #45438
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

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

1702 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2,3,4)(13,14)(15,21)(16,20)(17,18)(19,22)(23,27)(24,25)(26,28)(30,34)(31,37)(32,39)(42,44);(6,9)(10,12)(14,18)(15,22)(16,24)(26,27)(29,30,33,34)(31,35,37,41)(32,40,39,36)(38,44,43,42);(5,6,8,9)(7,10,11,12)(13,14)(15,19)(16,20)(17,18)(21,22)(23,26)(24,25)(27,28)(29,30)(31,35)(32,36)(33,34)(37,41)(38,42)(39,40)(43,44);(5,7,8,11)(6,10,9,12)(13,15)(14,19)(16,23)(17,22)(18,21)(20,26)(24,28)(25,27)(29,31,33,37)(30,35,34,41)(32,38,39,43)(36,42,40,44);(13,16,17,24)(14,20,18,25)(15,23,22,28)(19,26,21,27)(29,32,33,39)(30,36,34,40)(31,38,37,43)(35,42,41,44)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=fh,d^2=fgh,e^2=f,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dfg,e^b=ef

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, 464480);