#278641

order	512
nilpotency class	3
isoclinism	isoclinic class of #278635
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	#3106	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

522 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)(3,10,14,16)(4,9,12,7)(6,15,11,13)(17,18,23,26,22,28,21,27)(19,29,32,20,30,31,25,24);(1,2,4,7,5,8,12,9)(3,11,13,16,14,6,15,10)(17,19,21,25,22,30,23,32)(18,24,26,29,28,20,27,31);(1,2)(3,11)(4,7)(5,8)(6,14)(9,12)(10,15)(13,16)(17,20)(18,25)(19,27)(21,31)(22,24)(23,29)(26,30)(28,32);(1,3,12,15,5,14,4,13)(2,6,9,16,8,11,7,10)(17,19,23,32,22,30,21,25)(18,24,27,31,28,20,26,29)

Polycyclic presentation:

2^9|a^2=fi,b^2=e,d^2=eh,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,d^c=dh,f^c=fi,g^c=ghi

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, 185837);