#592606

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
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

1638 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)(7,11)(8,9)(10,12)(15,21)(19,24)(22,27)(26,28)(29,30,33,37)(31,35,39,34)(32,36,41,44)(38,43,40,42);(6,8)(7,12)(14,18)(19,24)(20,25)(26,28)(29,31,33,39)(30,34,37,35)(32,40,41,38)(36,43,44,42);(1,3,4,2)(5,6,9,8)(7,11,12,10)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28)(29,31)(30,35)(32,38)(33,39)(34,37)(36,43)(40,41)(42,44);(5,7,9,12)(6,10,8,11)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,30)(31,34)(32,36)(33,37)(35,39)(38,42)(40,43)(41,44);(5,8,9,6)(7,11,12,10)(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,32,33,41)(30,36,37,44)(31,38,39,40)(34,42,35,43)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=fi,d^2=fgh,e^2=h,b^a=bf,c^a=cg,d^a=dh,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 499802);