#197277

order	512
nilpotency class	3
isoclinism	isoclinic class of #197020
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	C₂⁴	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1026 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 104473);