#203154

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

678 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)(4,8)(5,10)(6,11)(7,12)(13,16)(14,15)(17,18,21,25,22,20,29,24)(19,23,28,31,27,26,32,30);(1,3,5,4,7,14,6,13)(2,8,11,9,12,16,10,15)(17,19,21,28,22,27,29,32)(18,23,25,31,20,26,24,30);(1,4,5,14,7,13,6,3)(2,9,11,16,12,15,10,8)(17,20)(18,22)(19,26)(21,24)(23,27)(25,29)(28,30)(31,32);(1,5,7,6)(2,10,12,11)(3,4,14,13)(8,15,16,9)(19,27)(23,26)(28,32)(30,31)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=e,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,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, 110350);