#198224

order	512
nilpotency class	3
isoclinism	isoclinic class of #198144
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	C₂×C₄²	C₂⁴
prime radical	C₂×#449	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1087 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 105420);