#503285

order	512
nilpotency class	4
isoclinism	isoclinic class of #501040
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

776 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

(5,6,9,12)(7,11,8,10)(13,14,16,19)(15,23,25,37)(17,30,28,43)(18,31,29,22)(20,34,32,26)(21,36,33,24)(27,42,41,39)(35,40,38,44);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(14,19)(20,32)(21,33)(22,31)(23,37)(30,43)(35,38)(39,42);(2,4)(13,15,24,40,18,27,28,34,16,25,36,44,29,41,17,26)(14,20,30,39,22,35,33,37,19,32,43,42,31,38,21,23);(5,8,9,7)(6,11,12,10)(14,19)(20,32)(21,33)(22,31)(23,37)(30,43)(35,38)(39,42)

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410481);