#501743

order	512
nilpotency class	4
isoclinism	isoclinic class of #501040
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

600 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)(13,14,19,33,26,20,27,40,17,22,18,23,15,24,34,37)(16,28,43,36,25,42,32,21,29,44,30,41,31,39,35,38);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15,17,26)(14,20,22,24)(16,25,29,31)(18,27,19,34)(21,36,38,41)(23,37,33,40)(28,39,44,42)(30,35,43,32);(2,4)(13,16,17,29)(14,21,22,38)(15,25,26,31)(18,32,19,35)(20,36,24,41)(23,39,33,42)(27,30,34,43)(28,37,44,40);(5,8,9,7)(6,11,12,10)(14,22)(20,24)(21,38)(23,33)(28,44)(36,41)(37,40)(39,42)

Polycyclic presentation:

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

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, 408939);