#582945

order	512
nilpotency class	2
isoclinism	isoclinic class of #52308
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

1818 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)(7,11)(10,12)(13,14,17,20)(15,18,23,21)(16,19,24,26)(22,25,28,27)(29,30,34,32)(31,37,39,35)(33,40,41,36)(38,42,44,43);(6,9)(10,12)(13,14,17,20)(15,21,23,18)(16,19,24,26)(22,27,28,25)(29,31,34,39)(30,35,32,37)(33,38,41,44)(36,42,40,43);(1,3,4,2)(5,6,8,9)(7,10,11,12)(13,14)(15,18)(16,19)(17,20)(21,23)(22,25)(24,26)(27,28)(29,31)(30,35)(32,37)(33,38)(34,39)(36,42)(40,43)(41,44);(5,7)(6,10)(8,11)(9,12)(13,15,17,23)(14,18,20,21)(16,22,24,28)(19,25,26,27)(29,32,34,30)(31,35,39,37)(33,40,41,36)(38,42,44,43);(13,16,17,24)(14,19,20,26)(15,22,23,28)(18,25,21,27)(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=g,c^2=hi,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=dgh,e^a=ef,c^b=ch,d^b=dg,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, 490141);