#278561

order	512
nilpotency class	3
isoclinism	isoclinic class of #278523
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	#3038	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

533 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)(3,10)(4,14)(5,8)(6,9)(7,11)(12,15)(13,16)(17,18,22,27,23,28,30,26)(19,29,21,24,25,20,31,32);(1,2,5,14,6,9,4,8)(3,7,12,16,13,15,11,10)(17,19,30,31,23,25,22,21)(18,24,27,20,28,32,26,29);(1,2)(3,7)(4,8)(5,14)(6,9)(10,11)(12,16)(13,15)(17,20,23,29)(18,25,28,19)(21,26,31,27)(22,24,30,32);(1,3,5,12,6,13,4,11)(2,7,14,16,9,15,8,10)(17,21,22,25,23,31,30,19)(18,20,26,24,28,29,27,32)

Polycyclic presentation:

2^9|a^2=f,b^2=eh,c^2=i,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,g^b=ghi,f^c=fi,g^c=ghi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 185757);