#199314

order	512
nilpotency class	3
isoclinism	isoclinic class of #199251
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	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60965	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1078 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:

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

Polycyclic presentation:

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

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, 106510);