#503180

order	512
nilpotency class	4
isoclinism	isoclinic class of #503179
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#25757
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

1128 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)(3,13)(4,19)(5,21)(6,23)(7,10)(8,12)(9,15)(11,22)(14,28)(16,27)(17,30)(18,29)(20,25)(24,31)(26,32);(1,3,6,15,8,18,22,17)(2,9,11,13,12,30,23,29)(4,20,10,26,21,27,31,28)(5,14,7,16,19,32,24,25);(1,4,7,23,6,10,19,2,8,21,24,11,22,31,5,12)(3,13,14,27,17,9,25,26,18,29,32,20,15,30,16,28);(1,5,22,24,8,19,6,7)(2,10,23,4,12,31,11,21)(3,14,17,25,18,32,15,16)(9,26,29,20,30,28,13,27)

Polycyclic presentation:

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

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, 410376);