#587136

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

1630 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)(5,6,10,12)(7,9,8,11)(15,21)(19,24)(22,27)(26,28)(29,30,33,36)(31,37,39,34)(32,40,41,35)(38,42,44,43);(5,7)(6,11)(8,10)(9,12)(14,18)(19,24)(20,25)(26,28)(29,30,33,36)(31,37,39,34)(32,35,41,40)(38,43,44,42);(6,12)(7,8)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28)(29,30)(31,34)(32,35)(33,36)(37,39)(38,42)(40,41)(43,44);(5,8,10,7)(6,9,12,11)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,31,33,39)(30,34,36,37)(32,38,41,44)(35,42,40,43);(5,9)(6,7)(8,12)(10,11)(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,32,33,41)(30,35,36,40)(31,38,39,44)(34,42,37,43)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=gh,d^2=h,e^2=fh,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, 494332);