#200005

order	512
nilpotency class	3
isoclinism	isoclinic class of #199842
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

1036 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(2,6)(3,10)(4,7)(9,14)(11,12)(13,16)(17,18,22,26,23,27,21,33)(19,24,29,38,30,39,35,37)(20,25,31,42,32,41,34,40)(28,36,45,46,44,47,43,48);(1,2,7,14,5,9,4,6)(3,8,10,16,11,15,12,13)(17,19,21,29,23,30,22,35)(18,24,26,37,27,39,33,38)(20,28,34,45,32,44,31,43)(25,36,42,48,41,47,40,46);(1,3)(2,8)(4,12)(5,11)(6,13)(7,10)(9,15)(14,16)(17,20)(18,25)(19,28)(21,34)(22,31)(23,32)(24,36)(26,42)(27,41)(29,45)(30,44)(33,40)(35,43)(37,48)(38,46)(39,47);(3,11)(8,15)(10,12)(13,16)(17,21,23,22)(18,26,27,33)(19,29,30,35)(20,31,32,34)(24,37,39,38)(25,40,41,42)(28,43,44,45)(36,46,47,48)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=i,e^2=hi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,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, 107201);