#196296

order	512
nilpotency class	3
isoclinism	isoclinic class of #196290
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	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1097 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 103492);