#348638

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	#3481	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

839 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)(3,9)(4,6)(5,13)(7,11)(8,12)(10,16)(14,15)(17,18,24,27)(19,23,26,22)(20,28,21,31)(25,29,32,30);(2,6)(7,15)(8,13)(9,16)(17,19,22,18,24,26,23,27)(20,28,29,25,21,31,30,32);(1,3)(2,7)(4,10)(5,14)(6,15)(8,16)(9,13)(11,12)(17,20)(18,25)(19,28)(21,24)(22,29)(23,30)(26,31)(27,32);(5,12)(8,13)(9,16)(11,14)(17,21)(18,25)(19,28)(20,24)(22,30)(23,29)(26,31)(27,32)

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,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, 255834);