#6445475

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437529
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3094 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

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

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, 6352671);