#501734

order	512
nilpotency class	4
isoclinism	isoclinic class of #501734
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24812
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3414	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

501 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408930);