#594162

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

1738 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)(15,18)(16,21)(17,22)(19,23)(20,25)(24,26)(27,28)(29,30,34,35)(31,36,37,32)(33,38,41,42)(39,43,44,40);(1,2,4,3)(6,9)(10,12)(14,15)(16,23)(17,25)(26,27)(30,35)(31,37)(33,41)(40,43);(1,3,4,2)(5,6,8,9)(7,10,11,12)(13,15,18,14)(16,19,23,21)(17,22,25,20)(24,26,28,27)(29,31)(30,36)(32,35)(33,39)(34,37)(38,43)(40,42)(41,44);(5,7)(6,10)(8,11)(9,12)(13,16,18,23)(14,19,15,21)(17,24,25,28)(20,26,22,27)(29,32,34,36)(30,37,35,31)(33,40,41,43)(38,44,42,39);(13,17)(14,20)(15,22)(16,24)(18,25)(19,26)(21,27)(23,28)(29,33,34,41)(30,38,35,42)(31,39,37,44)(32,40,36,43)

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=hi,d^2=f,e^2=fg,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, 501358);