#198490

order	512
nilpotency class	3
isoclinism	isoclinic class of #198171
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

1428 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=hi,e^2=hi,f^2=hi,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,d^b=di,e^b=ei,f^b=fhi,g^b=gi,d^c=dh,e^c=ehi,f^c=fhi,g^c=ghi

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, 105686);