#278611

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	C₂×#41	#41
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

503 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=i,d^2=e,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, 185807);