#6448116

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438949
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×#3038	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2270 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)(13,14,18,26,20,15,19,23)(16,24,17,22,28,21,27,25);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,20,14)(16,25,28,22)(17,21,27,24)(18,26,19,23);(2,4)(5,8,9,7)(6,10,11,12)(13,16,19,27,20,28,18,17)(14,21,23,22,15,24,26,25);(2,4)(13,17,19,16,20,27,18,28)(14,22,23,24,15,25,26,21);(13,18,20,19)(14,23,15,26)(16,17,28,27)(21,22,24,25)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6355312);