#503289

order	512
nilpotency class	4
isoclinism	isoclinic class of #501046
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

792 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ei,c^2=fh,d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410485);