#6448152

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2126 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:

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

Polycyclic presentation:

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

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, 6355348);