#289365

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

608 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=ef,d^2=eh,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, 196561);