#577005

order	512
nilpotency class	2
isoclinism	isoclinic class of #46350
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

1698 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)(14,18)(16,23)(19,24)(22,27)(29,30,33,34)(31,37,39,35)(32,40,42,36)(38,43,41,44);(5,6,8,10)(7,9,11,12)(15,21)(19,24)(22,27)(26,28)(30,34)(35,37)(36,40)(43,44);(5,6)(7,9)(8,10)(11,12)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28)(29,31,33,39)(30,35,34,37)(32,41,42,38)(36,44,40,43);(5,7,8,11)(6,9,10,12)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,30)(31,35)(32,36)(33,34)(37,39)(38,43)(40,42)(41,44);(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,32,33,42)(30,36,34,40)(31,38,39,41)(35,43,37,44)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=f,d^2=fgh,e^2=g,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dg,e^c=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, 484201);