#580036

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

1694 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=fg,d^2=hi,e^2=f,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, 487232);