#506477

order	512
nilpotency class	4
isoclinism	isoclinic class of #501040
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2186	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

837 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)(7,11)(8,10)(9,12)(13,14,17,21,29,36,43,44,18,22,31,38,16,20,28,35)(15,23,26,33,25,40,42,32,24,39,41,34,27,37,30,19);(1,2,3,4)(5,7,8,12)(6,9,10,11)(15,24)(19,32)(23,39)(25,27)(26,41)(30,42)(33,34)(37,40);(2,4)(13,15)(14,19)(16,25)(17,30)(18,24)(20,33)(21,37)(22,32)(23,35)(26,28)(27,29)(31,42)(34,36)(38,40)(39,44)(41,43);(6,10)(9,11)(13,16,18,29)(14,20,22,36)(15,25,24,27)(17,28,31,43)(19,33,32,34)(21,35,38,44)(23,40,39,37)(26,42,41,30)

Polycyclic presentation:

2^9|a^2=f,b^2=egi,d^2=i,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, 413673);