#505193

order	512
nilpotency class	4
isoclinism	isoclinic class of #505192
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#27243
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33531	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1765 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

641 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)(4,5)(6,10)(7,23)(8,12)(9,15)(11,25)(14,31)(16,28)(17,18)(19,22)(20,27)(21,26)(24,32)(29,30);(1,3,8,17)(2,9)(4,18)(5,15,6,21)(7,24,25,31)(10,13)(11,28)(12,26)(14,27)(16,20,29,22)(19,32)(23,30);(1,4,5,2,8,10,6,12)(3,14,15,30,17,32,21,28)(7,23,20,19,25,11,22,27)(9,24,13,29,26,31,18,16);(2,4,12,10)(7,25)(9,18,26,13)(11,27,23,19)(14,30,32,28)(16,29)(20,22)(24,31)

Polycyclic presentation:

2^9|b^2=fh,c^2=e,d^2=fhi,e^2=i,f^2=i,g^2=h,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,d^c=dh,g^c=ghi,h^c=hi,g^d=gi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 412389);