#200898

order	512
nilpotency class	3
isoclinism	isoclinic class of #200746
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	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1076 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 108094);