#6444607

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437521
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2040 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)(5,6,9,12)(7,11,8,10)(13,14)(15,18)(16,20)(17,23)(19,21)(22,26)(24,27)(25,28);(5,7,9,8)(6,10,12,11)(13,15,18,14)(16,24,26,28)(17,23,19,21)(20,25,22,27);(1,2,3,4)(5,7,9,8)(6,11,12,10)(13,16,18,26)(14,20,15,22)(17,25,19,27)(21,24,23,28);(5,8,9,7)(6,10,12,11)(13,16,18,26)(14,20,15,22)(17,25,19,27)(21,24,23,28);(13,17,18,19)(14,21,15,23)(16,25,26,27)(20,24,22,28)

Polycyclic presentation:

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

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, 6351803);