#461209

order	512
nilpotency class	3
isoclinism	isoclinic class of #461209
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16383
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

881 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 368405);