#427218

order	512
nilpotency class	3
isoclinism	isoclinic class of #417589
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1616 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=fi,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=dei,h^d=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, 334414);