#569987

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
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

1714 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)(5,6,9,8)(7,10,12,11)(14,18)(16,23)(19,24)(22,27)(29,30)(31,37)(32,40)(33,34)(35,38)(36,41)(39,43)(42,44);(5,6,9,8)(7,11,12,10)(14,18)(15,21)(20,25)(22,27)(30,33)(31,38)(32,41)(43,44);(1,2,4,3)(5,7,9,12)(6,10,8,11)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28)(29,31)(30,35)(32,42)(33,37)(34,38)(36,44)(39,41)(40,43);(1,3,4,2)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,32)(30,36)(31,39)(33,40)(34,41)(35,43)(37,44)(38,42);(5,8,9,6)(7,11,12,10)(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,33,34,30)(31,37,38,35)(32,40,41,36)(39,44,42,43)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=fgi,d^2=fhi,e^2=h,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, 477183);