#203107

order	512
nilpotency class	3
isoclinism	isoclinic class of #19420
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#388
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

637 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=e,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,c^b=ch,d^b=dhi,f^c=fhi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 110303);