#396555

order	512
nilpotency class	3
isoclinism	isoclinic class of #393928
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14734
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1848 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 303751);