#110729

order	512
nilpotency class	3
isoclinism	isoclinic class of #9331
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1437
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

592 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2)(3,9,11,16)(4,8)(5,7)(6,10,14,12)(13,15)(17,18,21,27,23,26,22,28)(19,29,32,41,31,24,35,38)(20,25,34,39,33,40,36,42)(30,37,45,47,44,43,46,48);(1,3)(2,6)(4,12)(5,11)(7,14)(8,16)(9,15)(10,13)(17,19,22,32,23,31,21,35)(18,24,28,38,26,29,27,41)(20,30,34,46,33,44,36,45)(25,37,39,48,40,43,42,47);(2,7)(4,13)(6,14)(10,12)(17,20,21,36,23,33,22,34)(18,25,28,39,26,40,27,42)(19,30,35,45,31,44,32,46)(24,37,38,48,29,43,41,47)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efi,d^2=i,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,d^b=di,e^b=ehi,f^b=fhi,d^c=dhi,e^c=eh,f^c=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17925);