#6444830

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#550	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2105 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

(2,4)(5,6)(7,10)(8,9)(11,12)(13,14,19,16)(15,20,26,27)(17,28,18,21)(22,25,23,24);(6,9)(10,12)(13,15,19,26)(14,20,16,27)(17,25,18,24)(21,22,28,23);(1,2,3,4)(13,14,19,16)(15,22,26,23)(17,21,18,28)(20,25,27,24);(13,16,19,14)(15,23,26,22)(17,28,18,21)(20,24,27,25);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,21,16,28)(15,24,26,25)(20,23,27,22)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gi,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6352026);