#589141

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, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1736 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=f,d^2=fh,e^2=gh,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, 496337);