#500741

order	512
nilpotency class	4
isoclinism	isoclinic class of #500740
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24236
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1049 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:

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

Polycyclic presentation:

2^9|b^2=e,d^2=ei,e^2=i,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,g^a=ghi,h^a=hi,c^b=cg,f^b=fhi,g^b=ghi,h^b=hi,f^c=fh,g^c=ghi,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, 407937);