#6440518

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	C₂²×#449	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3000 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^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, 6347714);