#193810

order	512
nilpotency class	3
isoclinism	isoclinic class of #32888
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	#2431	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1244 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,f^a=fh,e^b=eh,f^b=fh,g^b=gh,e^c=eh,g^c=gh

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, 101006);