#500867

order	512
nilpotency class	4
isoclinism	isoclinic class of #499891
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

916 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)(14,18)(15,21)(17,24)(19,27)(20,23)(22,28)(25,26);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,14,16,18)(15,19,22,26)(17,20,23,24)(21,25,28,27);(2,4)(13,15,16,22)(14,19,18,26)(17,25,23,27)(20,28,24,21);(5,8,9,7)(6,11,12,10)(13,14,16,18)(15,19,22,26)(17,20,23,24)(21,25,28,27)

Polycyclic presentation:

2^9|a^2=ei,b^2=e,c^2=i,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,f^c=fh,h^c=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, 408063);