#289388

order	512
nilpotency class	3
isoclinism	isoclinic class of #289363
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	#19673	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

569 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=ef,c^2=i,d^2=ei,e^2=h,f^2=i,g^2=h,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=gh

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, 196584);