#589126

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

1744 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)(5,6,9,10)(7,11,12,8)(14,18)(16,23)(19,24)(22,27)(29,30,34,37)(31,38,35,32)(33,41,42,36)(39,40,43,44);(1,3,4,2)(5,7,9,12)(6,8,10,11)(15,21)(19,24)(22,27)(26,28)(31,35)(32,38)(39,43)(40,44);(5,6,9,10)(7,11,12,8)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28)(29,31,34,35)(30,32,37,38)(33,39,42,43)(36,40,41,44);(5,6)(7,8)(9,10)(11,12)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,32)(30,35)(31,37)(33,40)(34,38)(36,43)(39,41)(42,44);(5,8)(6,7)(9,11)(10,12)(13,16)(14,20)(15,22)(17,23)(18,25)(19,26)(21,27)(24,28)(29,33,34,42)(30,36,37,41)(31,39,35,43)(32,40,38,44)

Polycyclic presentation:

2^9|a^2=i,b^2=fhi,c^2=fg,d^2=fgh,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, 496322);