#606375

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

1732 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)(5,6,10,8)(7,11,9,12)(14,18)(16,23)(19,24)(22,27)(29,30)(31,32)(33,41)(34,35)(36,39)(37,42)(38,43)(40,44);(5,7,10,9)(6,11,8,12)(15,21)(19,24)(22,27)(26,28)(30,35)(32,36)(33,42)(38,44);(1,2,4,3)(5,8,10,6)(7,11,9,12)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28)(29,31,34,39)(30,36,35,32)(33,38,42,44)(37,43,41,40);(1,2,4,3)(5,9,10,7)(6,11,8,12)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,32,34,36)(30,31,35,39)(33,40,42,43)(37,38,41,44);(1,3,4,2)(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,33)(30,37)(31,38)(32,40)(34,42)(35,41)(36,43)(39,44)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=fi,d^2=gi,e^2=fghi,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, 513571);