#289297

order	512
nilpotency class	3
isoclinism	isoclinic class of #289291
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3253	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

544 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:

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

Polycyclic presentation:

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

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, 196493);