#278623

order	512
nilpotency class	3
isoclinism	isoclinic class of #278603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#19484	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

540 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,d^2=ehi,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,g^b=ghi,f^c=fi,g^c=ghi

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, 185819);