#203134

order	512
nilpotency class	3
isoclinism	isoclinic class of #19279
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

690 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285591
maximal normal subgroup of	#72285591

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 110330);