#203104

order	512
nilpotency class	3
isoclinism	isoclinic class of #19384
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

657 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)(4,11)(7,14)(8,13)(12,15)(17,18,21,26,22,25,20,23)(19,27,29,24,30,32,28,31);(1,2,4,6,5,8,11,13)(3,7,10,12,9,15,16,14)(18,23,25,26)(24,27,31,32);(1,3,4,10,5,9,11,16)(2,7,6,12,8,15,13,14)(17,19,20,28,22,30,21,29)(18,24,26,32,25,31,23,27);(2,8)(6,13)(7,15)(12,14)(18,25)(23,26)(24,31)(27,32)

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,d^b=dhi,f^c=fhi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 110300);