#289300

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	#19332	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

558 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)(17,18,21,25,22,26,20,23)(19,27,29,32,30,31,28,24);(1,3,4,12,5,11,6,13)(2,7,9,14,10,15,8,16)(18,23,26,25)(24,31,32,27);(2,8)(3,11)(4,6)(7,14)(9,10)(15,16)(17,18)(19,24)(20,25)(21,23)(22,26)(27,28)(29,31)(30,32);(17,19,21,29,22,30,20,28)(18,24,23,31,26,32,25,27)

Polycyclic presentation:

2^9|a^2=fhi,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,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, 196496);