#198539

order	512
nilpotency class	3
isoclinism	isoclinic class of #198534
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×#3188	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1266 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=hi,e^2=hi,f^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,d^b=dhi,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fhi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 105735);