#198730

order	512
nilpotency class	3
isoclinism	isoclinic class of #198504
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1397 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=i,e^2=hi,f^2=hi,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fhi,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, 105926);