#348639

order	512
nilpotency class	3
isoclinism	isoclinic class of #348602
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3482	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

775 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,d^b=dh,f^c=fi,g^c=gi,f^d=fi,g^d=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, 255835);