#199968

order	512
nilpotency class	3
isoclinism	isoclinic class of #199815
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	#3444	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1017 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:

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

Polycyclic presentation:

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

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, 107164);