#502476

order	512
nilpotency class	4
isoclinism	isoclinic class of #502475
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#25384
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

669 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=ef,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,c^b=cg,g^b=gi,f^c=fh,g^c=ghi,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409672);