#198107

order	512
nilpotency class	3
isoclinism	isoclinic class of #19504
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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×#386
prime radical	#60942	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1130 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=i,e^2=hi,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,d^b=dhi,e^b=ei,f^b=fi,g^b=gi,d^c=dh,e^c=ei,f^c=fi,g^c=gi

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, 105303);