#289326

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	#3151	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

563 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=efh,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,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, 196522);