#507981

order	512
nilpotency class	4
isoclinism	isoclinic class of #501046
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

462 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,4,6)(3,7,8,5)(17,18,24,30,19,27,31,29,21,25,22,26,23,20,28,32);(1,3,4,8)(2,5,6,7)(9,10,12,13)(11,14,15,16)(17,19,21,23)(18,20,25,27)(22,28,24,31)(26,29,30,32);(10,13)(14,16)(17,20,21,27)(18,19,25,23)(22,30,24,26)(28,32,31,29);(2,6)(5,7)(9,11,12,15)(10,14,13,16)

Polycyclic presentation:

2^9|a^2=ef,b^2=eg,c^2=i,d^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,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, 415177);