#194309

order	512
nilpotency class	3
isoclinism	isoclinic class of #18682
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	#60949	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1141 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei,f^c=fi,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, 101505);