#4863179

order	512
nilpotency class	2
isoclinism	isoclinic class of #4861280
automorphism	Aut(G) order 2097152
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

1618 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fh,c^2=gh,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,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, 4770375);