#572107

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

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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=h,d^2=fgi,e^2=fhi,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, 479303);