#506492

order	512
nilpotency class	4
isoclinism	isoclinic class of #501064
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×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

795 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)(13,14,18,22,30,37,42,41,17,21,29,36,19,23,32,39)(15,16,25,27,40,33,44,43,24,20,34,35,26,28,38,31);(1,2,3,4)(5,7,8,12)(6,9,10,11)(13,15)(14,20)(16,21)(17,24)(18,25)(19,26)(22,35)(23,33)(27,36)(28,37)(29,34)(30,40)(31,41)(32,38)(39,43)(42,44);(2,4)(13,16,17,20)(14,15,21,24)(18,31,29,43)(19,33,30,28)(22,38,36,44)(23,40,37,26)(25,41,34,39)(27,42,35,32);(6,10)(9,11)(14,21)(15,24)(22,36)(23,37)(25,34)(26,40)(38,44)(39,41)

Polycyclic presentation:

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

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, 413688);