#6448115

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	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2116 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)(13,14,18,23,20,24,19,22)(15,25,26,28,27,17,21,16);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15)(14,16)(17,23)(18,21)(19,26)(20,27)(22,25)(24,28);(2,4)(5,8,9,7)(6,10,11,12)(13,16,18,25,20,28,19,17)(14,15,23,26,24,27,22,21);(2,4)(13,17,18,16,20,25,19,28)(14,21,23,15,24,26,22,27);(13,18,20,19)(14,22,24,23)(15,21,27,26)(16,25,28,17)

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=fg,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, 6355311);