#1634891

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1614 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=fhi,d^2=fg,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1542087);