#507293

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
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 ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

472 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)(9,10,12,13)(11,15,16,14)(17,18,21,23)(19,20,25,24)(22,26,27,31)(28,29,30,32);(1,3,4,8)(2,5,6,7)(10,13)(14,15)(17,19,27,30,18,24,31,29,21,25,22,28,23,20,26,32);(10,13)(14,15)(17,20,21,24)(18,25,23,19)(22,30,27,28)(26,32,31,29);(2,6)(5,7)(9,11,12,16)(10,14,13,15)(17,18,21,23)(19,20,25,24)(22,26,27,31)(28,29,30,32)

Polycyclic presentation:

2^9|a^2=e,b^2=efg,c^2=i,d^2=fi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^b=di,g^c=gh,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, 414489);