#6443803

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	#23393	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2242 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=fi,c^2=g,d^2=f,e^2=i,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, 6350999);