#496001

order	512
nilpotency class	4
isoclinism	isoclinic class of #2230
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#476
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₄×#476
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

542 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^2=egh,e^2=i,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,f^c=fhi,g^c=gi,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403197);