#198501

order	512
nilpotency class	3
isoclinism	isoclinic class of #198498
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	#60942	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1045 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 105697);