#5619973

order	512
nilpotency class	2
isoclinism	isoclinic class of #5605735
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1781 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=fgh,d^2=fgh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5527169);