#550418

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
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

1694 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)(13,14)(15,19)(16,24)(17,18)(20,25)(21,23)(22,27)(26,28)(29,30,34,32)(31,37,38,35)(33,40,41,36)(39,42,44,43);(1,2,4,3)(5,6,9,8)(7,11,12,10)(14,18)(19,21)(20,24)(26,27)(30,32)(31,38)(36,40)(39,44);(1,2,4,3)(5,7,9,12)(6,10,8,11)(13,14)(15,21)(16,20)(17,18)(19,23)(22,27)(24,25)(26,28)(29,31,34,38)(30,35,32,37)(33,39,41,44)(36,42,40,43);(5,6,9,8)(7,10,12,11)(13,15)(14,19)(16,22)(17,23)(18,21)(20,26)(24,27)(25,28)(29,32,34,30)(31,35,38,37)(33,40,41,36)(39,42,44,43);(1,3,4,2)(5,8,9,6)(7,11,12,10)(13,16,17,25)(14,20,18,24)(15,22,23,28)(19,26,21,27)(29,33,34,41)(30,36,32,40)(31,39,38,44)(35,42,37,43)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=i,d^2=fgh,e^2=fgi,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dg,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, 457614);