#194424

order	512
nilpotency class	3
isoclinism	isoclinic class of #1655
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	#60676	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1064 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)(3,12,13,16)(4,8,14,15)(5,9)(6,10)(7,11)(19,26,27,29)(23,30,32,31);(1,3,6,15,7,13,5,8)(2,4,10,16,11,14,9,12)(17,18)(19,23)(20,24)(21,28)(22,25)(26,31)(27,32)(29,30);(1,4)(2,8)(3,10)(5,12)(6,16)(7,14)(9,13)(11,15)(17,19)(18,23)(20,26)(21,29)(22,27)(24,31)(25,32)(28,30);(1,5,7,6)(2,9,11,10)(3,8,13,15)(4,12,14,16)(17,20,22,21)(18,24,25,28)(19,26,27,29)(23,31,32,30)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=i,e^2=h,f^2=hi,b^a=be,c^a=cf,f^a=fh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fhi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 101620);