#6439441

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×#3045	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2433 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)(5,6,8,9)(7,10,11,12)(13,14,17,21)(15,19,16,20)(18,22,27,28)(23,26,25,24);(6,9)(10,12)(13,14)(15,23)(16,25)(17,21)(18,28)(19,24)(20,26)(22,27);(2,4)(13,15,17,16)(14,19,21,20)(18,26,27,24)(22,25,28,23);(5,7,8,11)(6,10,9,12)(13,16,17,15)(14,20,21,19)(18,24,27,26)(22,23,28,25);(14,21)(19,20)(22,28)(23,25)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=f,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, 6346637);