#116141

order	512
nilpotency class	3
isoclinism	isoclinic class of #116141
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

661 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,e^2=h,f^2=hi,b^a=bd,c^a=ce,e^a=ei,c^b=cf,e^b=eh,f^b=fi,d^c=dh,e^c=eh,f^c=fhi,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 23337);