#198746

order	512
nilpotency class	3
isoclinism	isoclinic class of #198516
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

1300 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:

(1,2,5,9,6,10,4,7)(3,11,13,8,14,15,12,16)(17,18)(19,22,26,30)(20,27,28,23)(21,24)(25,29)(31,32);(2,7,10,9)(8,11,16,15)(17,19)(18,22,24,30)(20,25)(21,26)(23,29,27,32)(28,31);(1,3)(2,8)(4,12)(5,13)(6,14)(7,11)(9,15)(10,16)(17,20,21,28)(18,23,24,27)(19,25,26,31)(22,29,30,32);(19,26)(22,30)(25,31)(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,e^a=eh,f^a=fh,d^b=di,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, 105942);